Bibliography

A. General References

The books listed below are highly recommended for the quality of contents and presentation and their relevance to the subject matter of this book. This list is, of course, not exhaustive.

1909 LANDAU, E. Handbuch der Lehre der Verteilung der Primzahlen. Teubner, Leipzig, 1909. Reprinted by Chelsea, Bronx, N.Y., 1974. 1927 LANDAU, E. Vorlesungen uber Zahlentheorie (in 3 volumes). S. Hirzel, Leipzig, 1927. Reprinted by Chelsea, Bronx, N.Y., 1969. 1938 HARDY, G.H. and WRIGHT, E.M. An Introduction to the Theory of Numbers. Clarendon Press" Oxford, 1938 (fifth edition, 1979). 1952 DAVENPORT, H. The Higher Arithmetic. Hutchinson, London, 1952. Reprinted by Dover, New York, 1983. 1953 TROST, E. Primzahlen. Birkhauser, Basel, 1953 (second edition 1968). 1957 PRACHAR, K. Primzahlverteilung. Springer-Verlag, Berlin, 1957 (second edition 1978). 1962 SHANKS, D. Solved and Unsolved Problems in Number Theory. Spartan, Washington, 1962. Third edition by Chelsea, Bronx, N.Y., 1985. 1963 AYOUB, R.G. An Introduction to the Theory of Num­ bers. Amer. Math. Soc., Providence, R.I., 1963. 1964 SIERPINSKI, W. Elementary Theory of Numbers. Hafner, New York, 1964 (second edition, North-Holland, Amsterdam, 1987).

433 434 Bibliography

1974 HALBERSTAM, H. and RICHERT, H.E. Sieve Methods. Academic Press, New York, 1974. 1975 ELLISON, W.J. and MENDES-FRANCE, M. Les Nombres Premiers. Hermann, Paris, 1975. 1976 ADAMS, W.W. and GOLDSTEIN, L.J. Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, N.J., 1976. 1981 GUY, R.K. Unsolved Problems in Number Theory. Springer-Verlag, New York, 1981, second edition, 1994. 1982 HUA, L.K. Introduction to Number Theory. Springer­ Verlag, New York, 1982.

Besides the books listed above, I would like to mention expressly the following, just as well written, but with a fresh and distinct "regard" to the theory of numbers, in particular prime numbers.

1986 SCHROEDER, M.R. Number Theory in Science and Communication. Springer-Verlag, Berlin, 1986. 1994 CRANDALL, R.E. Projects in Scientific Computation. Springer-Verlag, New York, 1994.

B. Specific References

CHAPTER 1 1878 KUMMER, E.E. Neuer elementarer Beweis, dass die Anzahl aller Primzahlen eine unendliche ist. M onatsber. Akad. d. Wiss., Berlin, 1978/9, 777-778. 1890 STIELTJES, T.J. Sur la theorie des nombres. Etude bibliographique. An. Fac. Sci. Toulouse, 4, 1890, 1-103. 1891 HURWITZ, A. Ubungen zur Zahlen theorie, 1891-1918, (edited by H. Funk and B. Glaus). Bibl. E. T. H., Zurich, 1993. 1897 THUE, A. Mindre meddelelser II. Et be vis for at primtallenes antal er unendeligt. Arch. f Math. og N aturv., Kristiania, 19, No.4, 1897, 3-5. Reprinted in Selected Mathematical Papers (edited by T. Nagell, A. Selberg and S. Selberg), 31-32. Universitetsforlaget, Oslo, 1977. Chapter 1 435

1924 POLYA, G. and SZEGO, G. Aufgaben und Lehrsiitze aus der Analysis, 2 vols. Springer-Verlag, Berlin, 1924 (4th edition, 1970). 1947 BELLMAN, R. A note on relatively prime sequences. Bull. Amer. Math. Soc., 53,1947,778-779. 1955 FURSTENBERG, H. On the infinitude of primes. Amer. Math. Monthly, 62, 1955,353. 1959 GOLOMB, S.W. A connected topology for the integers. Amer. Math. Monthly, 66, 1959,663-665. 1963 MULLIN, A.A. Recursive function theory (A modem look on an Euclidean idea). Bull. Amer. Math. Soc., 69,1963, 737. 1964 EDWARDS, A.W.F. Infinite coprime sequences. Math. Gazette, 48, 1964,416-422. 1967 SAMUEL, P. Theorie Algebrique des N ombres. Hermann, Paris, 1967. English translation published by Houghton­ Miffiin, Boston, 1970. 1968 COX, C.D. and V AN DER POORTEN, A.J. On a sequence of prime numbers. J. Austr. Math. Soc. 8, 1968, 571-574. 1972 BORNING, A. Some results for k! + 1 and 2·3·5 ... p + 1. Math. Comp., 26, 1972,567-570. 1975 GUY, R.K. and NOWAKOWSKI, R. Discovering primes with Euclid. Delta, 5, 1975,49-63. 1978 MOHANTY, S.P. The number of primes is infinite. Fibonacci Quart., 16, 1978, 381-384. 1980 TEMPLER, M. On the primality of k! + 1 and 2 * 3 * ... * p + 1. Math. Comp., 34, 1980,303-304. 1980 WASHINGTON, L.c. The infinitude of primes via commutative algebra. Unpublished manuscript. 1982 BUHLER, J.P., CRANDALL, R.E., and PENK, M.A. Primes of the form n! ± 1 and 2· 3 ... p ± 1. Math. of Comp., 38, 1982, 639-643. 1985 DUBNER, H. and DUBNER, R. The development of a powerful low-cost computer for number theory applica­ tions. J. Recr. Math., 18(2), 1985/6,92-96. 1985 ODONI, R.W.K. On the prime divisors of the sequence wn+1 = 1 + Wi W2 ••• Wn• J. London Math. Soc., 32(2), 1985, 1-11. 436 Bibliography

1987 DUBNER, H. Factorial and primorial primes. J. Recr. Math., 19, 1987, 197-203. 1991 SHANKS, D. Euclid's primes. Bull. Inst. Comb. and Appl., 1, 1991, 33-36. 1993 CALDWELL, C. and DUBNER, H. Primorial, facto­ rial, and multifactorial primes. Math. Spectrum, 26, 1993/4, 1-7. 1993 WAGSTAFF Jr., S.S. Computing Eclid's primes. Bull. Inst. Comb. and Appl., 8, 1993,23-31. 1995 CALDWELL, C. On the primality of n! ± 1 and 2 x 3 x 5 x ... x p ± 1. Math. Comp., 64, 1995,889-890.

CHAPTER 2 1801 GAUSS, C.F. Disquisitiones Arithmeticae. G. Fleischer, Leipzig, 1801. English translation by A.A. Clarke. Yale University Press, New Haven, 1966. Revised English translation by W.e. Waterhouse. Springer-Verlag, New York, 1986. 1844 EISENSTEIN, F.G. Aufgaben. Journal fd. reine u. angew. Math., 27, 1844, 87. Reprinted in Mathematische Werke, Vol. I, 112. Chelsea, Bronx, N.Y., 1975. 1852 KUMMER, E.E. Uber die Erganzungssatze zu den allgemeinem Reciprocitatsgesetzen. Journal fd. reine u. angew. Math., 44, 1852, 93-146. Reprinted in Collected Papers, Vol. I (edited by A. Weil, 485-538). Springer­ Verlag, New York, 1975. 1876 LUCAS, E. Sur la recherche des grands nombres premiers. Assoc. Fram;aise p. I'Avanc. des Sciences, 5, 1876, 61-68. 1877 PEPIN, T. Sur la formule 22 " + 1. c.R. Acad. Sci. Paris, 85, 1877,329-331. 1878 LUCAS, E. Sur les congruences des nombres euleriens et des coefficients differentiels. Bull. Soc. Math. France, 6, 1878,49-54. 1878 LUCAS, E. Theorie des fonctions numeriques simple­ ment periodiques. Amer. J. Math., 1, 1878, 184-240 and 289-321. Chapter 2 437

1878 PROTH, F. Theoremes sur les nombres premiers. c.R. Acad. Sci. Paris, 85, 1877, 329-331. 1886 BANG, A.S. Taltheoretiske Unders0gelser. Tidskrift f Math., Ser. 5,4, 1886, 70-80 and 130-137. 1891 LUCAS, E. Theorie des N ombres. Gauthier-Villars, Paris, 1891. Reprinted by A. Blanchard, Paris, 1961. 1892 ZSIGMONDY, K. Zur Theorie der Potenzreste. M onatsh. f Math., 3, 1892, 265-284. 1899 KORSELT, A. Probleme chinois. L'Interm. Math., 6, 1899, 142-143. 1899 PERRIN, R. Item 1484, L'Interm. des l\1ath., 6, 1899, 76-77. 1903 MALO, E. Nombres qui, sans etre premiers, verifient exceptionnellement une congruence de Fermat. Llnterm. des Math., 10, 1903, 88. 1904 BIRKHOFF, G.D. and VANDIVER, H.S. On the integral divisors of an - bn. Annals of Math. (2), 5, 1904, 173-180. 1904 CIPOLLA, M. Sui numeri composti, P, che verificano la congruenza di Fermat aP - 1 == 1 (mod P). Annali di Matematica, (3), 9, 1904, 139-160. 1912 CARMICHAEL, R.D. On composite numbers P which satisfy the Fermat congruence aP - 1 == 1 (mod P). Amer. Math. Monthly, 19, 1912,22-27. 1913 CARMICHAEL, R.D. On the numeric all factors of the arithmetic forms ex n ± pn. Annals of Math., 15, 1913, 30- 70. 1913 DICKSON, L.E. Finiteness of odd perfect and primitive abundant numbers with n distinct prime factors. Amer. J. Math., 35, 1913, 413-422. Reprinted in The Collected Mathematical P apers (edited by A.A. Albert), V 01. I, 349-358. Chelsea, Bronx, N.Y., 1975. 1914 POCKLINGTON, H.C. The determination of the prime or composite nature of large numbers by Fer­ mat's theorem. Proc. Cambridge Phil. Soc., 18, 1914/6, 29- 30. 1921 PIRANDELLO, L. II Fu Mattia Pascal. Bemporad e Figlio, Firenze, 1921. 438 Bibliography

1922 CARMICHAEL, R.D. Note on Euler's ~-function. Bull. Amer. Math. Soc., 28, 1922, 109-110. 1925 CUNNINGHAM, A.J.e. and WOODALL, H.J. Fac­ torization of yn ± 1, y = 2, 3, 5,6, 7, 10, 11, 12 Up to High Powers (n). Hodgson, London, 1925. 1929 PILLAI, S.S. On some functions connected with ~(n). Bull. Amer. Math. Soc., 35, 1929,832-836. 1930 LEHMER, D.H. An extended theory of Lucas' func­ tions. Annals of Math., 31, 1930, 419-448. Reprinted in Selected Papers (edited by D. McCarthy), Vol. I, 11-48. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1932 LEHMER, D.H. On Euler's totient function. Bull. Amer. Math. Soc., 38, 1932, 745-757. Reprinted in Selected Papers (edited by D. McCarthy), Vol. I, 319-325. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1932 WESTERN, A.E. On Lucas' and Pepin's tests for the primeness of Mersenne's numbers. J. London Math. Soc., 7, 1932, 130-137. 1935 ARCHIBALD, R.C. Mersenne's numbers. Scripta Math., 3, 1935, 112-119. 1935 ERDOS, P. On the normal number of prime factors of p - 1 and some related problems concerning Euler's ~-function. Quart. J. Math. Oxford, 6, 1935,205-213. 1935 LEHMER, D.H. On Lucas' test for the primality of Mersenne numbers. J. London Math. Soc., 10, 1935, 162-165. Reprinted in Selected Papers (edited by D. McCarthy), Vol. I, 86-89. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1936 LEHMER, D.H. On the converse of Fermat's theorem. Amer. Math. Monthly, 43, 1936. Reprinted in Selected Papers (edited by D. McCarthy), Vol. I, 90-95. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1939 CHERNICK, J. On Fermat's simple theorem. Bull. Amer. Math. Soc., 45, 1939,269-274. 1941 KANOLD, H.J. Unterschungen tiber ungerade vol­ kommene Zahlen. Journal f. d. reine u. angew. Math., 183, 1941,98-109. Chapter 2 439

1942 KANOLD, H.J. Verscharfung einer notwendigen Be­ dingung fUr die Existenz einer ungeraden vollkommenen Zahl. Journal f d. reine u. angew. Math., 184, 1942, 116-123. 1943 BRAUER, A. On the nonexistence of odd perfect numbers of the form p(lqiq~ ... q?'-l qi. Bull. Amer. Math. Soc., 49, 1943, 712-718, and 937. 1944 KANOLD, H.J. Folgerungen aus dem Vorkommen einer Gauss' schen Primzahl in der Primfaktorenzerlegung einer ungeraden vollkommenen Zahl. Journal f d. reine u. angew. Math., 186,1944,25-29. 1944 PILLAI, S.S. On the smallest primitive root of a prime. J. Indian Math. Soc., (N.S.), 8,1944,14-17. 1944 SCHUH, F. Can n - 1 be divisible by ql(n} when n is composite? (in Dutch). Mathematica, Zutphen, B, 12, 1944, 102-107. 1945 KAPLANSKY, I. Lucas' tests for Mersenne numbers. Amer. Math. Monthly, 52, 1945, 188-190. 1947 KLEE, V.L. On a conjecture of Carmichael. Bull. Amer. Math. Soc., 53, 1947, 1183-1186. 1947 LEHMER, D.H. On the factors of 2n ± 1. Bull. Amer. Math. Soc., 53, 1947, 164-167. Reprinted in Selected Papers (edited by D. McCarthy), Vol. III, 1081-1084. Ch. Baggage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1947 LEVIT, R.J. The nonexistence of a certain type of odd perfect number. Bull. Amer. Math. Soc., 53, 1947, 392-396. 1948 ORE, O. On the averages of the divisors of a number. Amer. Math. Monthly, 55, 1948,615-619. 1948 STEUERWALD, R. Uber die Kongruenz an- 1 == 1 (mod n). Sitzungsber. math.-naturw. K1. Bayer. Akad. Wiss. Miinchen, 1948,69-70. 1949 BERNHARD, H.A. On the least possible odd perfect number. Amer. Math. Monthly, 56, 1949,628-629. 1949 ERDOS, P. On the converse of Fermat's theorem. Amer. Math. Monthly, 56, 1949,623-624. 1949 FRIDLENDER, V.R. On the least nth power non­ residue (in Russian). Doklady Akad. Nauk SSSR (N.S.), 66, 1949, 351-352. 440 Bibliography

1949 KUHNEL, U. Verscharfung der notwendigen Bedin~ gungen fUr die Existenz von ungeraden vollkommenen Zahlen. Math. Z., 52, 1949,202-211. 1949 SHAPIRO, H.N. Note on a theorem of Dickson. Bull. Amer. Math. Soc., 55, 1949,450-452. 1950 BEEGER, N.G.W.H. On composite numbers n for which an- 1 == 1 (mod n), for every a, prime to n. Scripta Math., 16, 1950, 133-135. 1950 GIUGA, G. Su una presumibile proprieta caratteristica dei numeri primi. 1st. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat., (3), 14 (83), 1950,511-528. 1950 GUPTA, H. On a problem of Erdos. Amer. Math. Monthly, 57, 1950,326-329. 1950 KANOLD, H.J. Satze tiber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretische Probleme, II. Journal! d. reine u. angew. Math., 187, 1950,355-366. 1950 SALlE, H. Uber den kleinsten positiven quadratischen Nichtrest einer Primzahl. Math. Nachr., 3, 1949, 7-8. 1950 SOMAYAJULU, B.S.K.R. On Euler's totient function ¢J(n). Math. Student, 18, 1950,31-32. 1951 BEEGER, N.G.W.H. On even numbers m dividing 2m - 2. Amer. Math. Monthly, 58, 1951,553-555. 1951 MORROW, D.e. Some properties of D numbers. Amer. Math. Monthly, 58, 1951,329-330. 1951 WEBBER, G.e. Nonexistence of odd perfect numbers of the form 32P . prJ.. SiP1S~P2S~P3. Duke Math. J., 18, 1951, 741-749. 1952 DUPARC, H.J.A. On Carmichael numbers. Simon Stevin, 29, 1952,21-24. 1952 GRUN, O. Uber ungerade vollkommene Zahlen. Math. Zeits., 55, 1952,353-354. 1953 KANOLD, H.J. Einige neuere Bedingungen fUr die Existenz ungerader vollkommener Zahlen. Journal ! d. reine u. angew. Math., 192, 1953,24-34. 1953 KNODEL, W. Carmichaelsche Zahlen. Math. Nachr., 9, 1953,343-350. 1953 SELFRIDGE, J.L. Factors of Fermat numbers. Math. Comp., 7, 1953,274-275. Chapter 2 441

1953 TOUCHARD, J. On prime numbers: and perfect numbers. Scripta Math., 19, 1953,35-39. 1954 KANOLD, H.J. Uber die Dichten der Mengen der vollkommenen und der befreundeten Zahlen. Math. Zeits., 61, 1954, 180-185. 1954 ROBINSON, R.M. Mersenne and Fermat numbers. Proc. Amer. Math. Soc., 5, 1954,842-846. 1954 SCHINZEL, A. Quelques theon!mes sur les fonctions ¢J(n) et O"(n). Bull. Acad. Polon. Sci., Cl. III, 2, 1954, 467- 469. 1954 SCHINZEL, A. Generalization of a theorem of B.S.K.R. Somayajulu on the Euler's function ¢J(n). Ganita, 5, 1954, 123-128. 1954 SCHINZEL, A. and SIERPINSKI, W. Sur quelques proprietes des fonctions ¢J(n) et 0" (n). Bull Acad. Polon. Sci., CI. III, 2, 1954,463-466. 1955 ARTIN, E. The orders of the linear groups. Comm. Pure and Appl. Math., 8, 1955, 355-365. Reprinted in Collected Papers (edited by S. Lang and 1.T. Tate), 387-397. Addison-Wesley, Reading, Mass., 1965. 1955 HORNFECK, B. Zur Dichte der Menge der vollkom­ menen Zahlen. Arch. Math., 6, 1955,442-443. 1955 LABORDE, P. A note on the even perfect numbers. Amer. Math. Monthly, 62, 1955, 348-349. 1955 WARD, M. The intrinsic divisors of Lehmer numbers. Annals of Math., (2), 62, 1955,230-236. 1956 HORNFECK, B. Bemerkung zu meiner Note tiber vollkommene Zahlen. Arch. Math., 7, 1956,273. 1956 KANOLD, H.J. Uber einen Satz von L.E. Dickson, II. Math. Ann., 132, 1956,246-255. 1956 ORE, O. Number Theory and Its History. McGraw-Hill, New York, 2nd edition, 1956. 1956 SCHINZEL, A. Sur l'equation ¢J(x) = m. Elem. Math., 11, 1956, 75-78. 1956 SCHINZEL, A. Sur un probleme concernant la fonc­ tion ¢J(n). Czechoslovak Math. J., 6, (81), 1956, 164-165. 1956 SIERPINSKI, W. Sur une propriete de Ia fonction ¢J(n). Publ. Math. Debrecen, 4, 1956, 184-185. 442 Bibliography

1957 HORNFECK, B. and WIRSING, E. Uber die Haufig­ keit vollkommener Zahlen. Math. Ann., 133, 1957, 431- 438. 1957 KANOLD, H.J. Uber die Verteilung der vollkommenen Zahlen und allgemeinerer Zahlenmengen. Math. Ann., 132, 1957,442-450. 1957 KANOLD, H.J. Uber mehrfach vollkommene Zahlen, II. Journal! d. reine u. angew. Math., 197, 1957,82-96. 1957 McCARTHY, P.J. Odd perfect numbers. Scripta Math., 23, 1957,43-47. 1957 ROBINSON, R.M. The converse of Fermat's theorem. Amer. Math. Monthly., 64, 1957, 703-710. 1958 ERDOS, P. Some remarks on Euler's ¢-function. Acta Arithm., 4., 1958, 10-19. 1958 JARDEN, D. Recurring Sequences. Riveon Lemate­ matika, Jerusalem, 1958 (3rd edition, revised by J. Brillhart, Fibonacci Assoc, San Jose, 1973). 1958 PERISASTRI, M. A note on odd perfect numbers. Math. Student, 26, 1958, 179-181. 1958 ROBINSON, R.M. A report on primes of the form k·2n + 1 and on factors of Fermat numbers. Proc. Amer. Math. Soc., 9, 1958,673-681. 1958 SCHINZEL, A. Sur l'equation ¢(x + k) = ¢(x). Acta Arithm., 4, 1958, 181-184. 1958 SIERPINSKI, W. Sur les nombres premiers de la forme nn + 1. L'Enseign. Math., (2), 4, 1958,211-212. 1959 DURST, L.K. Exceptional real Lehmer sequences. Pacific J. Math., 9, 1959,437-441. 1959 ROTKIEWICZ, A. Sur les nombres composes qui divisent an- 1 - bn- 1• Rend. Circ. Mat. Palermo, (2), 8, 1959, 115-116. 1959 ROTKIEWICZ, A. Sur les nombres pairs a pour lesquels les nombres anb - abn, respectivement an- 1 - bn- 1 sont divisibles par n. Rend. Circ. Mat. Palermo, (2), 8, 1959, 341-342. 1959 SATYANARAYANA, M. Odd perfect numbers. Math. Student, 27, 1959, 17-18. Chapter 2 443

1959 SCHINZEL, A. Sur les nombres composes n qui divisent an - a. Rend. Circ. Mat. Palermo, (2) 7,1958, 1-5. 1959 SCHINZEL, A. and W AKULICZ, A. Sur l'equation ¢J(x + k) = ¢J(x), II. Acta Arithm., 5, 1959, 425-426. 1959 WARD, M. Tests for primality based on Sylvester's cyclotomic numbers. Pacific J. Math., 9, 1959, 1269-1272. 1959 WIRSING, E. Bemerkung zu der Arbeit iiber vollkom­ mene Zahlen. Math. Ann., 137, 1959,316-318. 1960 INKERI, K. Tests for primality. Annales Acad. Sci. Fennicae, Ser. A, I, 279, Helsinki, 1960, 19 pages. Collected Papers of Kustaa Inkeri (edite par T. MetsankyIa et P. Ribenboim), Queen's Papers in Pure and Appl. Math., 91, 1992, Queen's Univ., Kingston, Ontario, Canada. 1961 ERDOS, P. Remarks in number theory, I (in Hungar­ ian, with English summary). Mat. Lapok, 12, 1961, 10-17. 1961 NORTON, K.K. Remarks on the number of factors of an odd perfect number. Acta Arithm., 6, 1961, 365-374. 1961 WARD, M. The prime divisors of Fibonacci numbers. Pacific J. Math., 11, 1961,379-389. 1962 BURGESS, D.A. On character sums and L-series. Proc. London Math. Soc., (3), 12, 1962, 193-206. 1962 CROCKER, R. A theorem on pseudo-primes. Amer. Math. Monthly, 69, 1962,540. 1962 HURWITZ, A. New Mersenne primes. Math. Comp., 16, 1962,249-251. 1962 M";'KOWSKI, A. Generalization of Morrow's D num­ bers. Simon Stevin, 36, 1962, 71. 1962 SCHINZEL, A. The intrinsic divisors of Lehmer numbers in the case of negative discriminant. Arkiv. for Mat., 4, 1962,413-416. 1962 SCHINZEL, A. On primitive prime factors of an - bn. Proc. Cambridge Phil. Soc., 58, 1962,555-562. 1962 SHANKS, D. Solved and Unsolved Problems in Number Theory. Spartan, Washington, 1962. 3rd edition by Chelsea, Bronx, N.Y., 1985. 1963 BATEMAN, P.T. Solution of a problem by Ore. Amer. Math. Monthly, 70, 1963, 101-102. 444 Bibliography

1963 ORE, O. Euler's function problem 4995 [1961, 1010]. Amer. Math. Monthly, 70, 1963, 101. 1963 SCHINZEL, A. On primitive prime factors of Lehmer numbers, I. Acta Arithm., 8, 1963, 211-223. 1963 SCHINZEL, A. On primitive prime factors of Lehmer numbers, II. Acta Arithm., 8,1963,251-257. 1963 SELFRIDGE, J.L. Solution of a problem by Ore. Amer. Math. Monthly, 70, 1963, 101. 1963 SURYANARA YANA, D. On odd perfect numbers, II. Proc. Amer. Math., 14, 1963,896-904. 1964 BIERMANN, K.-R. Thomas Clausen. Mathematiker und Astronom, Journal J.d. reine u. angew. Math. 216, 1964, 159-198. 1964 GILLIES, D.B. Three new Mersenne primes and a statistical theory. Math. Comp., 18, 1964,93-98. 1964 LEHMER, E. On the infinitude of Fibonacci pseudo­ primes. Fibonacci Quart., 2, 1964,229-230. 1964 SELFRIDGE, J.L. and HURWITZ, A. Fermat num­ bers and Mersenne numbers, Math. Comp., 18, 1964, 146-148. 1965 ERDOS, P. Some recent advances and current prob­ lems in number theory, in Lectures in Modern Mathemat­ ics, Vol. III (edited by T.L. Saaty), 196-244. Wiley, New York, 1965. 1965 ROTKIEWICZ, A. Sur les nombres de Mersenne depourvus de facteurs carres et sur les nombres naturels n tels que n2 12n - 2. Matem. Vesnik (Beograd), 2(17), 1965, 78-80. 1966 GROSSWALD, E. Topics from the Theory of Numbers. Macmillan, New York, 1966; second edition Birkhauser, Boston, 1984. 1966 MUSKAT, J.B. On divisors of odd perfect numbers. Math. Comp., 20, 1966, 141-144. 1967 BRILLHART, J. and SELFRIDGE, J.L. Some fac­ torizations of 2m ± 1 and related results. Math. Comp., 21, 1967,87-96 and corrigendum page 751. 1967 MOZZOCHI, c.J. A simple proof of the Chinese remainder theorem. Amer. Math. Monthly, 74, 1967,998. Chapter 2 445

1967 TUCKERMAN, B. Odd perfect numbers: A search procedure, and a new lower bound of 10 36• IBM Res. Rep., RC-1925, 1967, and Notices Amer. Math. Soc., 15, 1968,226. 1968 SCHINZEL, A. On primitive prime factors of Lehmer numbers, III. Acta Arithm., 15, 1968,49-69. 1968 WILLIAMS, H.C. and ZARNKE, C.R. A report on prime numbers of the forms M = (6a + 1) X 22m- 1 - 1 and M' = (6a - 1) X 22m - 1. Math. Comp., 22,1968,420-422. 1969 LEHMER, D.H. Computer technology applied to the theory of numbers. In Studies in Number Theory (edited by W.l. Le Veque), 117-151. Math. Assoc. of America, Washington, 1969. 1970 ELLIOTT, P.D.T.A. On the mean value of f(p). Proc. London Math. Soc., (3), 21, 1970, 28-96. 1970 LIEUWENS, E. Do there exist composite numbers for which k¢J(M) = M - 1 holds? Nieuw. Arch. Wisk., (3), 18,1970, 165-169. 1970 PARBERRY, E.A. On primes and pseudo-primes related to the Fibonacci sequence. Fibonacci Quart., 8, 1970,49-60. 1970 SURYANARAY ANA, D. and HAGIS., Jr., P. A theorem concerning odd perfect numbers. Fibonnaci Quart., 8, 1970,337-346 and 374. 1971 LIEUWENS, E. Fermat Pseudo-Primes. Ph.D. Thesis, Delft, 1971. 1971 MORRISON, M.A. and BRILLHART, J. The factori­ zation of F7 • Bull. Amer. Math. Soc., 77, 1971,264. 1971 SCHONHAGE, A. and STRASSEN, V. Schneller Muitiplikation grosser Zahlen. Computing, 7, 1971, 281-292. 1971 TUCKERMAN, B. The 24th Mersenne prime. Proc. Nat. Acad. Sci. U.S.A., 68, 1971,2319-2320. 1972 HAGIS, Jr., P. and McDANIEL, W.L. A new result concerning the structure of odd perfect numbers. Proc. Amer. Math. Soc., 32,1972,13-15. 1972 MANN, H.B. and SHANKS, D. A necessary and sufficient condition for primality and its source. J. Comb. Th., ser. A, 13,1972,131-134. 446 Bibliography

1972 MILLS, W.H. On a conjecture of Ore. Proc. 1972 Number Th. Con! in Boulder, pp. 142-146. 1972 RIBENBOIM, P. Algebraic Numbers. Wiley-Interscience, New York, 1972. 1972 ROBBINS, N. The nonexistence of odd perfect num­ bers with less than seven distinct prime factors. Notices Amer. Math. Soc., 19, 1972, A52. 1972 ROTKIEWICZ, A. Pseudoprime Numbers and their Generalizations. Stud. Assoc. Fac. Sci. Univ. Novi Sad, 1972. 1972 ROTKIEWICZ, A. Un probleme sur les nombres pseudo-premiers. Indag. Math., 34, 1972,86-91.

1972 WILLIAMS, H.C. The primality of N = 2A3 11 - 1. Can. Math. Bull., (4),15, 1972,585-589. 1973 HAGIS, Jr., P. A lower bound for the set of odd perfect numbers. Math. Comp., 27, 1973,951-953. 1973 HONSBERGER, R. Mathematical Gems, I. Math. Assoc. of America, Washington, 1973. 1973 ROTKIEWICZ, A. On pseudoprimes with respect to the Lucas sequences. Bull. Acad. Polon. Sci., 21, 1973, 793-797. 1974 LIGH, S. and NEAL, L. A note on Mersenne numbers. Math. Mag., 47, 1974,231-233. 1974 POLLARD, J.M. Theorems on factorization and pri­ mality testing. Proc. Cambridge Phil. Soc., 76, 1974, 521-528. 1974 POMERANCE, C. On Carmichael's conjecture. Proc. Amer. Math. Soc., 43, 1974,297-298. 1974 SCHINZEL, A. Primitive divisors of the expression A" - B" in algebraic number fields. Journal! d. reine u. angew. Math., 268/269, 1974,27-33. 1974 SINHA, T.N. Note on perfect numbers. Math. Student, 42, 1974, 336. 1975 BRILLHART, J., LEHMER, D.H. and SELFRIDGE, J.L. New primality criteria and factorizations of 2m ± 1. Math. Comp., 29, 1975,620-647. 1975 HAGIS, Jr., P. and McDANIEL, W.L. On the largest prime divisor of an odd perfect number, II. Math. Comp., 29, 1975,922-924. Chapter 2 447

1975 MORRISON, M.A. A note on primality testing using Lucas sequences. Math. Comp., 29,1975, 181-182. 1975 MORRISON, M.A. and BRILLHART, J. A method of factoring and the factorization of F7 • Math. Comp., 29, 1975, 183-208. 1975 POLLARD, J.M. A Monte-Carlo method for factoriza­ tion. BIT, 15, 1975,331-334. 1975 POMERANCE, C. The second largest prime factor of an odd perfect number. Math. Comp., 29, 1975, 914- 921. 1975 PRATT, V.R. Every prime has a succinct certificate. SIAM J. Comput., 4, 1975,214-220. 1975 STEWART, c.L. The greatest prime factor of an - bn. Acta Arithm., 26, 1975,427-433. 1976 BUXTON, M. and ELMORE, S. An extension of lower bounds for odd perfect numbers. Notices Amer. Math. Soc., 23, 1976, A55. 1976 DIFFIE, W. and HELLMAN, M.E. New directions in crypto~raphy. IEEE Trans. on Inf. Th., IT-22. 1976 ERDOS, P. and SHOREY, T.N. On the greatest prime factor of 2P - 1 for a prime p, and other expressions. Acta Arithm., 30, 1976,257-265. 1976 GUY, R.K. How to factor a number. Proc. Fifth Manitoba Conf Numerical Math., 1976,49-89 (Congressus Numerantium, XVI, Winnipeg, Manitoba, 1976). 1976 LEHMER, D.H. Strong Carmichael numbers. J. Aus­ tral. Math. Soc., A, 21, 1976, 508-510. Reprinted in Selected Papers (edited by D. McCarthy), Vol. I, 140-142. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1976 MENDELSOHN, N.S. The equation fjJ(x) = k. Math. Mag., 49, 1976,37-39. 1976 MILLER, G.L. Riemann's hypothesis and tests for primality. J. Compo Syst. Sci., 13, 1976,300-317. 1976 RABIN, M.O. Probabilistic algorithms. In Algorithms and Complexity (edited by J.F. Traub), 21-39. Academic Press, New York, 1976. 448 Bibliography

1976 RECAMAN SANTOS, B. Twelve and its totatives. Math. Mag., 49, 1970,239-240. 1976 TRAUB, J.F. (Editor) Algorithms and Complexity. Academic Press, New York, 1976. 1976 YAO, A.C. On the evaluation of powers. SIAM J. Comput., 5, 1976, 100-103. 1976 YORINAGA, M. On a congruential property of Fibo­ nacci numbers. Numerical experiments. Considerations and remarks. Math. J. Okayama Univ., 19, 1976, 5-10; 11-17. 1977 AIGNER, A. Das quadratfreie Kern der Eulerschen fj}-Funktion. Monatsh. Math., 83, 1977,89-91. 1977 GARDNER, M. Mathematical Games: A new kind of cypher that would take millions of years to break. Scientific American, August 1977, 120-124. 1977 KISHORE, M. Odd perfect numbers not divisible by 3 are divisible by at least ten distinct primes. Math. Comp., 31, 1977, 274-279. 1977 KISHORE, M. The Number of Distinct Prime Factors for Which a(N) = 2N, a(N) = 2N ± 1 and fj}(N)IN - 1. Ph.D. Thesis, University of Toledo, Ohio, 1977, 39 pages. 1977 MALM, D.E.G. On Monte-Carlo primality tests. Notices Amer. Math. Soc., 24, 1977, A-529, abstract 77T-A22. 1977 POMERANCE, C. On composite n for which fj}(n)ln - 1. II Pacific J. Math., 69, 1977, 177-186. 1977 POMERANCE, C. Multiply perfect numbers, Mersenne primes and effective computability. Math. Ann., 226, 1977, 195-206. 1977 SOLOVAY, R. and STRASSEN, V. A fast Monte-Carlo test for primality. SIAM J. Comput., 6, 1977, 84-85. 1977 STEWART, C.L. On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers. Proc. London Math. Soc., (3), 35, 1977, 425-447. 1977 STEWART, c.L. Primitive divisors of Lucas and Lehmer numbers. In Transcendence Theory: Advances and Applications (edited by A. Baker and D.W. Masser), 79-92. Academic Press, London, 1977. Chapter 2 449

1977 WILLIAMS, H.C. On numbers analogous to the Carmichael numbers. Can. Math. Bull., 20, 1977, 133-143. 1978 COHEN, G.L. On odd perfect numbers. Fibonacci Quart., 16, 1978, 523-527. 1978 HAUSMAN, M. and SHAPIRO, H.N. Adding tota­ tives. Math. Mag., 51, 1978,284-288. 1978 KISS, P. and PHONG, B.M. On a function concerning second order recurrences. Ann. Univ. Sci. Budapest, 21, 1978, 119-122. 1978 RIVEST, R.L. Remarks on a proposed cryptanalytic attack on the M.I.T. public-key cryptosystem. Crypto[ogia, 2, 1978, 62-65. 1978 RIVEST, R.L., SHAMIR, A. and AD LEMAN, L.M. A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM, 21, 1978, 120-126. 1978 SLOWINSKI, D. Searching for the 27th Mersenne prime. J. Rech. Math., 11 (4), 1978,258-261. 1978 WILLIAMS, H.C. Primality testing on a computer. Ars Comb., 5, 1978, 127-185. 1978 YORINAGA, M. Numerical computation of Carmichael numbers. Math. J. Okayama Univ., 20, 1978, 151-163. 1979 CHEIN, E.Z. Non-existence of odd perfect numbers of the form q'i 1q'22 ... q(,6 and 5a1 q'22 ... qg9. Ph.D. Thesis, Pennsylvania State Univ., 1979. 1979 LENSTRA Jr., H.W. Miller's primality test. Inf Proc. Letters, 8(2), 1979,86-88. 1979 PLAISTED, D.A. Fast verification, testing, and genera­ tion oflarge primes. Theoret. Compo Sci., 9, 1979, 1-16. 1979 YORINAGA, M. Numerical computation of Carmichael numbers, II. Math. J. Okayama Univ., 21, 1979, 183-205. 1980 BAILLIE, R. and WAGSTAFF, Jr., S.S. Lucas pseudo­ primes. Math. Comp., 35,19801391-1417. 1980 COHEN, G.L. and HAGIS, Jr., P. On the number of prime factors of n if ~(n)l(n - 1). Nieuw Arch. Wisk., (3), 28, 1980, 1770-185. 1980 HAGIS, Jr., P. Outline of a proof that every odd perfect number has at least eight prime factors. Math. Comp., 35, 1980, 1027-1032. 450 Bibliography

1980 MONIER, L. Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoret. Comput. Sci., 12, 1980, 97-108. 1980 NOLL, L.C. and NICKEL, L. The 25th and 26th Mersenne numbers. Math. Comp., 35,1980,1387-1390. 1980 POMERANCE, c., SELFRIDGE, J.L., and WAG­ STAFF, Jr., S.S. The pseudoprimes to 25.109 . Math. Comp., 35, 1980, 1003-1026. 1980 RABIN, M.O. Probabilistic algorithm for testing primality. J. Nb. Th., 12, 1980, 128-138. 1980 WAGSTAFF, Jr., S.S. Large Carmichael numbers. Math. J. Okayama Univ., 22, 1980,33-41. 1980 WALL, D.W. Conditions for r/J(N) to properly divide N - 1. In A Collection of Manuscripts Related to the Fibonacci Sequence (edited by YE. Hoggatt and M. Bicknell-Johnson), 205-208. Eighteenth Anniv. Vol., Fibonacci Assoc., San Jose, 1980. 1980 YORINAGA, M. Carmichael numbers with many prime factors. Math. J. Okayama Univ., 22, 1980, 169- 184. 1981 BRENT, R.P. and POLLARD, J.M. Factorization of the eighth Fermat number. Math. Comp., 36, 1981, 627-630. 1981 DIXON, J.D. Asympototically fast factorization of integers. Math. Comp., 36, 1981,255-260. 1981 GROSSW ALD, E. On Burgess' bound for primitive roots modulo primes and an application to r(p). Amer. J. Math., 103, 1981, 1171-1183. 1981 KISHORE, M. On odd perfect, quasi perfect and odd almost perfect numbers. Math. Comp., 36, 1981,583-586. 1981 KNUTH, D.E. The Art of Computer Programming, Vol. 2, 2nd edition. Addison-Wesley, Reading, Mass., 1981. 1981 KONHEIM, A.G. Cryptography: A Primer. Wiley­ Interscience, New York, 1981. 1981 LENSTRA, Jr., H.W. Primality testing algorithms (after Adleman, Rumely and Williams). In Seminaire Bourbaki, expose No. 576. Lecture Notes in Math., # 901, 243-257. Springer-Verlag, Berlin, 1981. Chapter 2 451

1981 LUNEBURG, H. Ein einfacher Beweis fUr den Satz von Zsigmondy tiber primitive Primteiler von AN - 1. In Geometries and Groups (edited by M. Aigner and D. Jungnickel). Lecture Notes in Math., # 893, 219-222. Springer-Verlag, New York, 1981. 1981 POMERANCE, C. Recent developments in primality testing. Math. Intelligencer, 3, No.3, 1981,97-105. 1981 SHOREY, T.N. and STEWART, C.L. On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers, II. J. London Math. Soc., (2), 23, 1981, 17-23. 1982 ADAMS, W. and SHANKS, D. Strong primality tests that are not sufficient, Math. Comp., 35, 1983,255-300. 1982 BRENT, R.P. Succinct proofs of primality for the factors of some Fermat numbers. Math. Comp., 38, 1982, 253-255. 1982 COUVREUR, C. and QUISQUATER, J.J. An intro­ duction to fast generation of large prime numbers. Philips J. Res., 37,1982,231-264. 1982 HOOGENDOORN, P.J. On a secure public-key cryptosystem. In Computational Methods in Number Theory (edited by H.W. Lenstra, Jr. and R. Tijdeman), Part I, 159-168. Math. Centre Tracts, # 154, Amsterdam, 1982. 1982 KENDALL, D.G. The scale of perfection. J. Appl. Prob., 19A, 1982, 125-138. 1982 LENSTRA, Jr., H.W. Primality testing. In Computa­ tional Methods in Number Theory (edited by H.W. Lenstra, Jr. and R. Tijdeman), Part I, 55-77. Math. Centre Tracts, # 154, Amsterdam, 1982. 1982 MASAI, P. and VALETTE, A. A lower bound for a counterexample in Carmichael's conjecture. Boll. Un. Mat. Ital., (6), I-A, 1982,313-316. 1982 NAUR, T. Integer Factorization. DAIMI PB-144, Aarhus University, 1982, 129 pages. 1982 POMERANCE, C. The search for prime numbers. Scient. Amer., 247,1982, 122-131. 1982 POMERANCE, C. Analysis and comparison of some integer factoring algorithms. In Computational Methods in Number Theory (edited by H.W. Lenstra, Jr. and R. 452 Bibliography

Tijdeman), Part I, 89-139. Math. Centre Tracts, # 154, Amsterdam, 1982. 1982 VAN EMDE BOAS, P. Machine models, computa­ tional complexity and number theory. In Computational Methods in Number Theory (edited by H.W. Lenstra, Jr. and R. Tijdeman), Part I, 7-42. Math. Centre Tracts, # 154, Amsterdam, 1982. 1982 VORHOEVE, M. Factorizations algorithms of expo­ nential order. In Computational Methods in Number Theory (edited by H.W. Lenstra, Jr. and R. Tijdeman), Part I, 79-87. Math. Centre Tracts, # 154, Amsterdam, 1982. 1982 WILLIAMS, H.C. A class of primality tests for trinomials which includes the Lucas-Lehmer test. Pacific J. Math., 98,1982,477-494. 1982 WILLIAMS, H.C. The influence of computers in the development of number theory. Compo and Maths. with Appl., 8, 1982, 75-93. 1982 WOODS, D. and HUENEMANN, J. Larger Canmchael numbers. Comput. Math. and Appl., 8,1982,215-216. 1983 ADLEMAN, L.M., POMERANCE, C. and RUMELY, R.S. On distinguishing prime numbers from composite numbers. Annals of Math., (2), 117, 1983, 173-206. 1983 BRILLHART, J., LEHMER, D.H., SELFRIDGE, J.L., TUCKERMAN, B., and WAGSTAFF, Jr., S.S. Factori­ zations of bn ± 1, b = 2, 3, 5 ,6, 7, 10, 11, 12 up to High Powers. Contemporary Math., Vol. 22, Amer. Math. Soc., Providence, R.I., 1983 (second edition 1988). 1983 DA VIS, J.A. and HOLDRIDGE, D.B. Factorization using the quadratic sieve algorithm. Tech. Rep. SAND 83-1346, Sandia Nat. Lab., Albuquerque (1983). 1983 GERVER, J.L. Factoring large numbers with a quadratic sieve. Math. Comp., 36, 1983,287-294. 1983 HAGIS, Jr., P. Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors. Math. Comp., 40, 1983,399-404. 1983 KELLER, W. Factors of Fermat numbers and large primes of the form k· 2n + 1. Math. Comp., 41, 1983, 661-673. Chapter 2 453

1983 KISHORE, M. Odd perfect numbers not divisible by 3, II. Math. Comp., 40, 1983,405-411. 1983 NAUR, T. New integer factorizations. Math. Comp., 41, 1983,687-695. 1983 POMERANCE, e. and WAGSTAFF, Jr., S.S. Imple­ mentation of the continued fraction integer factoring algorithm. Congressus Numerantium, 37,1983,99-118. 1983 POWELL, B. Problem 6420 (On primitive roots). Amer. Math. Monthly, 90, 1983,60. 1983 RUMELY, R.S. Recent advances in primality testing. Notices Amer. Math. Soc., 30, No.5, 1983,475-477. 1983 SINGMASTER, D. Some Lucas pseudloprimes. Ab­ stracts Amer. Math. Soc., 4, 1983, No. 83T-1O-146, 197. 1983 WUNDERLICH, M.e. A performance analysis of a simple prime-testing algorithm, Math. Comp., 40, 1983, 709-714. 1983 YATES, S. Titanic primes. J. Recr. Math., 16, 1983/4, 250-260. 1984 COHEN, H. and LENSTRA, Jr., H.W. Primality testing and Jacobi sums. Math. Comp., 42, 1984,297-330. 1984 DIXON, J.D. Factorization and primality tests. Amer. Math. Monthly, 91, 1984,333-352. 1984 GORDON, J. Strong primes are easy to find. Proc. Eurocrypt 84, 1984,216-223. Springer-Verlag, New York. 1984 KEARNES, K. Solution of problem 6420. Amer. Math. Monthly, 91, 1984, 521. 1984 NICOLAS, J.L. Tests de primalite. Expo. Math., 2, 1984,223-234. 1984 POMERANCE, e. Lecture Notes on Primality Testing and Factoring (notes by G.M. Gagola Jr.). Math. Assoc. America, Notes, No.4, 1984,34 pages. 1984 ROTKIEWICZ, A. On the congruence 2n - 2 == 1 (mod n). Math. Comp., 43, 1984,271-272. 1984 WILLIAMS, H.C. An overview of factoring. In Ad­ vances in Cryptology (edited by D. Chaum), 71-80. Plenum, New York, 1984. 1984 WILLIAMS, H.e. Factoring on a computer. Math. Intelligencer, 6, 1984, No.3, 29-36. 454 Bibliography

1984 SCHOOF, R.J. Quadratic fields and factorization. In Computational Methods in Number Theory, (editors H.W. Lenstra, Jr. and R. Tijdeman), Math. Centre Tracts 154/ 155, Amsterdam, 1984,235-286. 1984 YATES, S. Sinkers of the Titanics. J. Recr. Math., 17, 1984/5,268-274. 1985 BACH, E. Analytic Methods in the Analysis and Design of Number- Theoretic Algorithms. A.C.M. Distinguished Dissertations. M.I.T. Press, Cambridge, 1985. 1985 BEDOCCHI, E. Note on a conjecture on prime numbers. Rev. Mat. Univ. Parma (4),11, 1985,229-236. 1985 BOSMA, W. Primality testing using elliptic curves. Math. Inst., Univ. Amsterdam, Report 85-12, 54 pages, 1985. 1985 CHUDNOVSKY, D.V. and CHUDNOVSKY, G.V. Se­ quences of numbers generated by addition in formal groups and new primality and factorization tests. Research Report RC-11262, IBM, Yorktown Heights, NY, 1985. 1985 DUBNER, H. and DUBNER, R. The development of a powerful low-cost computer for number theory applica­ tions. J. Recr. Math., 18, 1985/6,92-96. 1985 KELLER, W. The 17th prime of the form 5· 2n + 1. Abstract Amer. Math. Soc., 6, No.1, 1985, 121. 1985 ONDREJKA, R. Ten extraordinary primes. J. Recr. Math., 18, 1985/86,87-92. 1985 RIESEL, H. Prime Numbers and Computer Methodsfor Factorization. Birkhauser, Boston, 1985 (second edition 1994). 1985 SCHOOF, R.J. Elliptic curves over finite fields and the computation of square roots modulo p. Math. Comp., 44, 1985,483-494. 1985 SOMER, L. The divisibility and modular properties of kth order linear recurrences over the ring of the integers on an algebraic number field. Ph.D. Thesis. University of Illinois, Urbana-Champaign, 1985. 1985 WAGON, S. Perfect numbers. Math. Intelligencer, 7, No.2, 1985, 66-68. Chapter 2 455

1986 AD LEMAN, L.M. and HUANG, M.-D.A. Recognizing primes in random polynomial time. Preprint, 1986. 1986 ATKIN, A.O.L. Manuscript of a lecture in Boulder, CO, 1986. 1986 ERDOS, P. and POMERANCE, C. On the number of false witnesses for a composite number. Math. Comp., 46, 1986,259-279. 1986 GOLDWASSER, S. and KILIAN, J. Almost all primes can be quickly certified. Proc. 18th Annual ACM Symp. on Theory of Computing, Berkeley, 1986,316-219. 1986 KISS, P., PHONG, H.M. and LIEUWENS, E. On Lucas pesudoprimes which are products of s primes. In Fibonacci Numbers and their Applications (edited by A.N. Philippou, G.E. Bergum, and A.F. Horadam), 131-139. Reidel, Dordrecht, 1986. 1986 LENSTRA, Jr., H.W. Elliptic curves and number­ theoretic algorithms. Math. Inst., Univ. Amsterdam, Report 86-19, 18 pages, 1986. 1986 LENSTRA, Jr., H.W. Factoring integers with elliptic curves. Math. Inst., Univ. Amsterdam. Report 86-18, 19 pages, 1986. 1986 SHEN, M.K. On the congruence 2n- k == 1 (mod n). Math. Comp., 46, 1986, 715-716. 1986 SILVERMAN, J.H. The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986. 1986 WAGON, S. Carmichael's "Empirical Theorem." Math. Intelligencer, 8, No.2, 1986, 61-62. 1986 WAGON, S. Primality testing. Math. Intelligencer, 8, No.3, 1986, 58-61. 1987 AD LEMAN, L.M. and HUANG, M.-D.A. Recognizing primes in a random polynomial time. Proc. 19th Annual ACM Symposium on Theory of Computing, 1987, 462- 469. 1987 AD LEMAN, L.M. and McCURLEY, K.S. Open prob­ lems in number theoretic complexity, in Discrete Algo­ rithms and Complexity, 237-262. Academic Press, New York,1987. 456 Bibliography

1987 COHEN, G.L. On the largest component of an odd perfect number. J. Austral. Math. Soc., (A), 42, 1987, 280-286. 1987 COHEN, H. and LENSTRA, A.K. Implementation of a new primality test (and Supplement). Math. Comp., 48, 1987, 103-121, and S1-S4. 1987 KISS, P. and PHONG, B.M. On a problem of A. Rotkiewicz. Math. Comp., 48,1987,751-755. 1987 KOBLITZ, N. A Course in Number Theory and Cryptography. Springer-Verlag, New York, 1987. 1987 LENSTRA, Jr., H.W. Factoring integers with elliptic curves. Annals Math., 126, 1987,649-673. 1987 LI YAN and DU SHIRAN Chinese Mathematics, A Concise History (English translation by Crossley, J.N. and Lun, A.W.c.). Clarendon Press, Oxford, 1987. 1987 McDANIEL, W.L. The generalized pseudoprime con­ gruence an- k == bn- k (mod n). c.R. Math. Rep. Acad. Sci. Canada, 9, 1987, 143-148. 1987 POMERANCE, C. Very short primality proofs. Math. Comp., 48, 1987,315-322. 1988 BEAUCHEMIN, P., BRASSARD, G., CREPEAU, c., and POMERANCE, C. The generation of random numbers that are probably prime. J. Cryptology, 1, 1988, 53-64. 1988 BRILLHART, J., MONTGOMERY, P.L., and SILVER­ MAN, R.D. Tables of Fibonacci and Lucas factoriza­ tions (and Supplement). Math. Comp., 50, 1988, 251-260 and S1-S15. 1988 DUBNER, H. Prime triplets in an arithmetic progres­ sion starting with 3. J. Recr. Math., 20, 1988,211-213. 1988 MORAIN, F. Implementation of the Atkin-Goldwasser­ Kilian primality testing algorithm, Rapport de Recherche 911, INRIA, Octobre 1988. 1988 PHONG, B.M. Generalized solution of Rotkiewicz's problem. Mat. Lapok, to appear in 1988(?). 1988 YOUNG, J. and BUELL, D.A. The twentieth Fermat number is composite. Math. Comp., 50, 1988,261-262. Chapter 2 457

1989 BACH, E. and SHALLIT, J. Factoring with cyclotomic polynomials. Math. Comp., 52, 1989,201-209. 1989 BATEMAN, P.T., SELFRIDGE, J.L., and WAGSTAFF, Jr., S.S. The new Mersenne conjecture. Amer. Math. Monthly, 96, 1989, 125-128. 1989 BRESSOUD, D.M. Factorization and Primality Testing. Springer-Verlag, New York, 1989. 1989 BRENT, R.P. Factorization of the eleventh Fermat number (preliminary report), Abstracts Amer. Math. Soc., 10, 1989,89 T-l1-73. 1989 BRENT, R.P. and COHEN, G.L. A new lower bound for odd perfect numbers. Math. Comp., 53, 1989,431-437 and S-7 to S-24. 1989 DUBNER, H. A new method for producing large Carmichael numbers. Math. Comp., 53, 1989,411-414. 1989 GORDON, D.M. On the number of el1iptic pseudo­ primes. Math. Comp., 52, 1989,231-245. 1989 GORDON, D.M. Pseudoprimes on elliptic curves, in Number Theory. Proc. Intern. Conf. on Number Theory, Univ. Laval, 1987 (editors 1.M. de Koniack and C. Levesque), 290-305. W. de Gruyter, Berlin, 1989. 1989 LEMOS, M. Criptograjia, Numeros Primos e Algo­ ritmos. (17 0 Col6quio Brasileiro de Matematica) Inst. Mat. Pura e Aplic., Rio de 1aneiro, 1989, 72 pages. 1989 PINTZ, J., STEIGER, W.L., and SZEMEREDI, E. Infinite sets of primes with fast primality tests and quick certification of large primes. Math. Comp., 53, 1989, 399-406. 1989 SOMER, L. On Fermat d-pseudoprimes. In Theorie des Nombres, Number Theory (editors 1.M. de Koninck and C. Levesque), W. de Gruyter, Berlin, 1989,841-860. 1989 TZANAKIS, N. and De WEGER, B.M.M. On the practical solution of Thue equation. J. Nb. Th., 31, 1989, 99-132. 1990 DUBNER, H. and NELSON, H. Carmichael numbers which are the product of three Carmichael numbers. J. Recr. Math., 22, 1990,2-6. 458 Bibliography

1990 GURAK, S. Pseudoprimes for higher order linear recurrence sequences. Math. Comp., 55, 1990, 783-813. 1990 LENSTRA, A.K. and LENSTRA, Jr., H.W. Algorithms in number theory. Handbook of Theoretical Computer Science (edited by Van Leeuwen, J., Meyer, A., Nivat, M., Paterson, M., and Perrin, D.). North-Holland, Amsterdam, 1990. 1990 LENSTRA, A.K. and MANASSE, M.S. Factoring by electronic mail. Advances in Cryptology, Eurocrypt '89, Lect. Notes in Compo Sci., 434, 1990, 355-37l. 1990 LENSTRA, A.K., LENSTRA, Jr., H.W., MANASSE, M.S., and POLLARD, J.M. The number field sieve. Proc. 22nd Annual A.CM. Symp. Th. Compo (STOC), Baltimore, 1990, 564-572. 1990 MAURER, U.M. Fast generation of secure RSA products with almost maximal diversity. Advances in Cryptology-EUROCRYPT '89. Lecture Notes in Com­ puter Science, 434, 1990, 636-647. Springer-Verlag, New York. 1990 McDANIEL, W.L. The existence of solutions of the generalized pseudoprime congruence af(n) == bf(n) (mod n). Colloq. Math., 49,1990,177-190. 1990 MONTGOMERY, P.L. and SILVERMAN, R.D. An FFT extension to the P-1 factoring algorithm. Math. Comp., 54, 1990, 839-854. 1990 MORAIN, F. Atkin's test: news from the front. Advances in Cryptology-EUROCRYPT '89. Lecture Notes in Computer Science, 434, 1990, 626-635. Springer-Verlag, New York. 1990 VAN LEEUWEN, J. (Ed.) Handbook of Theoretical Computer Science. A. Algorithms and Complexity. North­ Holland, Amsterdam, 1990. 1991 ARNO, S. A note on Perrin pseudoprimes. Math. Comp., 56, 1991,371-376. 1991 COLQUITT, W.N. and WELSH Jr., L. A new Mersenne prime. Math. Comp., 56, 1991,867-870. 1991 FRASNAY, C. Extension a l'anneau 7Lp du theon!me de Lucas, sur les coefficients binomiaux. Singularite, 2, 1991, 13-15. Chapter 2 459

1991 MORAIN, F. Distributed primality proving and the primality of (2 2529 + 1)/3, Advances in Cryptology­ EUROCRYPT '90 (Proc. Workshop on the Theory and Appl. of Cryptographic Techniques, Aarhus, Denmark. May 21-24, 1990), (I. B. Damgard, ed.), Lecture Notes in Comput. Sci., vol. 473, Springer-Verlag, Berlin and New York, 1991, pp. 110-123. 1992 ADLEMAN, L.M. and HUANG, M.-D.A. Primality Testing and Abelian Varieties. Springer Lect. Notes in Math. # 1512, New York, 1992. 1992 GUILLAUME, D. and MORAIN, F. Building Car­ michael numbers with a large number of prime factors and generalization to other numbers. Research report, Ec. Poly technique, Palaiseau, 1992. 1992 KELLER, W. Factors of Fermat numbers and large primes of the form k x 2n + 1, II. Preprint, Univ. Hamburg (1992). 1992 KELLER, W. and MORAIN, F. The complete fac­ torization of some large Mersenne composites. Abstracts Amer. Math. Soc., 13,1992,506. 1992 MORAIN, F. Prime values of partition numbers and the primality of P1840926. Rep. Res. No. 11, Ecole Poly technique, Palaiseau, 1992. 1992 MORAIN, F. Elliptic curves, primality proving and some titanic primes. Journees Arithmetiques 1989 Aster­ is que, vol. 198-199-200, Soc. Math. France, Paris, 1992, 245-251. 1992 YATES, S. Collecting gigantic and titanic primes. J. Recr. Math., 24, 1992, 187-195. 1992 ZHANG, M. Searching for large Carmichael numbers. Sichuan Daxue Ziebao, 29, 1992,472-474. 1993 ATKIN, A.O.L. and MORAIN, F. Elliptic curves and primality proving. Math. Comp., 61, 1993,29-58. 1993 BRENT, R.P., COHEN, G.L. and TE RIELE, H.J.J. Improved techniques for lower bounds for odd perfect numbers. Math. Comp., 61,1993,857-868. 1993 COHEN, H. A Course in Computational Algebraic Number Theory. Springer-Verlag. New York, 1993. 460 Bibliography

1993 JAESCHKE, G. On strong pseudoprimes to several bases. Math. Comp., 61, 1993, 915-926. 1993 LENSTRA, A.K., LENSTRA Jr., H.W., MANASSE, M.S., and POLLARD, J.M. The factorization of the ninth Fermat number. Math. Comp., 61, 1993, 319- 349. 1993 LENSTRA, A.K., LENSTRA Jr., H.W., MANASSE, J.M., and POLLARD, J.M. The number field sieve, in The Development of Number Field Sieve (Lenstra, A.K. and Lenstra, Jr., H.W., editors). Lecture Notes in Mathematics, # 1554,11-42. Springer-Verlag, New York, 1993. 1993 PINCH, R.G.E. The Carmichael numbers up to 10 15. Math. Comp., 61, 1993, 703-722. 1993 POMERANCE, G. Carmichael numbers. Nieuw Arch. v. Wiskunde, (4) 11, 1993, 199-209. 1993 SILVERMAN, R.D. and WAGSTAFF Jr., S.S. A practical analysis of the elliptic curve factoring algorithm. Math. Compo 61, 1993,445-462. 1993 WILLIAMS, H.C. How was F6 factored. Math. Comp., 61, 1993,463-474. 1994 ADLEMAN, L.M. and McCURLEY, K.S. Open problems in number theoretic complexity, II. In Algo­ rithmic Number Theory (First International Symposium A.N.T.S., I, Ithaca, N.Y., May 1994). Lecture Notes in Computer Science, 877, 291-302. Springer-Verlag, New York, 1994. 1994 ALFORD, W.R., GRANVILLE, A. and POMERANCE, C. There are infinitely many Carmichael numbers. Annals of Math., 140, 1994, 703-722. 1994 BORWEIN, D., BORWEIN, J.M., BORWEIN, P.B., and GURGENSOHN, R. Giuga's conjecture on pri­ mality. (To appear in Amer. Math. Monthly). 1994 HEATH-BROWN, D.R. Odd perfect numbers. Math. Proc. Cambridge Phil. Soc., 115,1994,191-196. 1994 LENSTRA, A.K. Factoring. In Proceedings of the Workshop on Distributed Algorithms on Graphs, Lect. Notes in Compo Sci. #857, 28-38. Springer-Verlag, New York, 1994. Chapter 3 461

1994 LOH, G. and NIEBUHR, W. A new algorithm for constructing large Carmichael numbers (to appear in Math. Camp.). 1994 SCHLAFLY, A. and WAGON, S. Carmichael's conjec­ ture on the Euler function is valid below 1010000000. Math. Camp., 63, 1994,415-419. 1994 SCHOR, P.W. Algorithms in quantum computation: Discrete log and factoring. DIMACS, Techn. Rep. 94-37 to appear in Proc. 35th Symp. on Found. Compo Sci. (1994). 1994 SILVERMAN, J.H. and TATE, J. Rational Points on Elliptic Curves. Springer-Verlag, New York, 1994. 1995 CALDWELL, C. On the primality of n! ± 1 and 2·3·5··· p ± 1. Math. Comp., 64,1995,889-890. 1995 CRANDALL, R.E., DOENIAS, J., NORRIE, c., and YOUNG, J. The twenty-second Fermat number is composite. Math. Comp., 64, 1995,863-868. 1995 DUBNER, H. and KELLER, W. Factors of generalized Fermat numbers. Math. Comp., 64, 1995,397-405. 1995 GOSTIN, G.B. New factors of Fermat numbers. Math. Comp., 64, 1995,393-395. 1995 RIBENBOIM, P. The Fibonacci numbers and the Arctic Ocean (Proceedings of the Second Gaussian Symposium, Miinchen, 1994), Symposia Gaussian, Conf. A (editors M. Behara, R. Fritsch, R.G. Lintz), vv. de Gruyter, Berlin, 1995,41-83. 1995 RIBENBOIM, P. Selling primes. Math Afag., 68, 1995, 175-182. 1995 TREVISAN, V. and CARVALHO, J.B. The composite character of the twenty-second Fermat number. J. Supercomp., 9, 1995, 179-182. 1995 VOUTIER, P.M. Primitive divisors of Lucas and Lehmer sequences. Math. Comp., 64, 1995,869-888.

CHAPTER 3 1839/1840 DIRICHLET, G.L. Recherches sur diverses appli­ cations de l'analyse infinitesimale a la theorie des nombres. 462 Bibliography

Journal! d. reine u. angew. Math., 19, 1839,342-369 and 21, 1840, 1-12 and 134-135. Reprinted in Werke (edited by L. Kronecker), Vol. I, 411-496. G. Reimer, Berlin, 1889. Reprinted by Chelsea, Bronx, N.Y., 1969. 1852 KUMMER, E.E. Uber die Erganzungssatze zu den allgemeinem Reciprocitatsgestzen. Journal ! d. reine u. angew. Math., 44, 1852, 93-146. Reprinted in Collected Papers (edited by A. Weil), Vol. 1,485-538. Springer-Verlag, New York, 1975. 1912 FROBENIUS, F.G. Uber quadratische Formen, die viele Primzahlen darstellen. Sitzungsber. d. Konigl. Akad. d. Wiss. zu Berlin, 1912, 966-980. Rerinted in Gesammelte Abhandlungen, vol. III, 573-587. Springer-Verlag, Berlin, 1968. 1912 RABINOVITCH, G. Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkorpern. Proc. Fifth Intern. Congress Math., Cambridge, Vol. 1, 1912, 418-421. 1918 LANDAU, E. Uber die Klassenzahl imaginar quadra­ tischer Zahlkorper. Gottinger Nachr., 1918,285-295. 1933 DEURING, M. Imaginar-quadratische Zahlkorper mit Klassenzahl Eins. Invent. Math., 5,1968,169-179. 1933 LEHMER, D.H. On imaginary quadratic fields whose class number is unity. Bull. Amer. Math. Soc., 39, 1933,360. 1934 HEILBRONN, H. On the class number in imaginary quadratic fields. Quart. J. Pure and Appl. Math., Oxford, Ser. 2, 5, 1934, 150-160. 1934 HEILBRONN, H. and LINFOOT, E.H. On the imaginary quadratic corpora of class-number one. Quart. J. Pure and Appl. Math., Oxford, Ser. 2,5, 1934,293-301. 1934 MORDELL, L.J. On the Riemann hypothesis and imaginary quadratic fields with a given class number. J. London Math. Soc., 9, 1934,289-298. 1935 SIEGEL, c.L. Uber die Classenzahl quadrati scher Zahlkorper. Acta Arithm., 1, 1935, 83-86. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan and H. MaaB), Vol. I, 406-409. Springer-Verlag, Berlin, 1966. Chapter 3 463

1936 LEHMER, D.H. On the function x 2 + x + A. Sphinx, 6, 1936,212-214. 1938 SKOLEM, T. Diophantische Gleichung'en. Springer­ Verlag, Berlin, 1938. 1943 REINER, I. Functions not formulas for primes. Amer. Math. Monthly, 50, 1943,619-621. 1946 BUCK, R.C. Prime-representing functions. Amer. Math. Monthly, 53, 1946,265. 1947 MILLS, W.H. A prime-representing function. Bull. Amer. Math. Soc., 53,604. 1951 WRIGHT, E.M. A prime-representing function. Amer. Math. Monthly, 58, 1951,616-618. 1952 HEEGNER, K. Diophantische Analysis und Modul­ funktionen. Math. Zeits., 56, 1952,227-253. 1952 SIERPINSKI, W. Sur une formule donnant to us les nombres premiers. c.R. Acad. Sci. Paris, 235, 1952, 1078-1079. 1960 PUTNAM, H. An unsolvable problem in number theory. J. Symb. Logic, 1960,220-232. 1962 COHN, H. Advanced Number Theory. Wiley, New York, 1962. Reprinted by Dover, New York, 1980. 1964 BOREVICH, Z.I. and SHAFAREVICH, I.R. Number Theory. Moscow, 1964. English translation by N. Greenleaf, published by Academic Press, New York, 1966. 1964 WILLANS, c.P. On formulae for the nth prime. Math. Gaz., 48, 1964,413-415. 1966 BAKER, A. Linear forms in the logarithms of algebraic numbers. Mathematika, 13, 1966,204-216. 1967 GOODSTEIN, R.L. and WORMELL, c.P. Formulae for primes. Math. Gaz., 51, 1967 35-38. 1967 STARK, H.M. A complete determination of the complex quadratic fields of class-number one. Michigan Math. J., 14, 1967, 1-27. 1968 DEURING, M. Imaginare quadratische Zahlkorper mit der Klassenzahl Eins. Invent. Math., 5, 1968, 169-179. 1969 BAKER, A. A remark on the class number of quadratic fields. Bull. London Math. Soc., 1, 1969,98-102. 464 Bibliography

1969 DUDLEY, U. History of a formula for primes. Amer. Math. Monthly, 76, 1969,23-28. 1969 STARK, H.M. A historical note on complex quadratic fields with class-number one. Proc. Amer. Math. Soc., 21, 1969,254-255. 1971 BAKER, A. Imaginary quadratic fields with class number 2. Annals of Math., 94,1971,139-152. 1971 BAKER, A. On the class number of imaginary quadratic fields. Bull. Amer. Math. Soc., 77,1971,678-684. 1971 GANDHI, J.M. Formulae for the nth prime. Proc. Washington State U niv. Conf. on Number Theory, 96-106. Wash. St. Univ., Pullman, Wash., 1971. 1971 MATIJASEVIC, Yu. V. Diophantine representation of the set of prime numbers (in Russian). Dokl. Akad. N auk SSSR, 196, 1971, 770-773. English translation by R.N. Goss, in Soviet Math. Dokl. 12, 1971,354-358. 1972 RIBENBOIM, P. Algebraic Numbers. Wiley-Interscience, New York, 1972. 1972 SIEGEL, c.L. Zur Theorie der quadratischen Formen. Nachr. Akad. Wiss. Gottingen, Math. Phys. Kl., 1972, No. 3, 21-46. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan and H. MaaB), Vol. IV, 224-249. Springer-Verlag, New York, 1979. 1972 VANDEN EYNDEN, C. A proof of Gandhi's formula for the nth prime. Amer. Math. Monthly, 79, 1972,625. 1973 DAVIS, M. Hilbert's tenth problem is unsolvable. Amer. Math. Monthly, 80,1973,233-269. 1973 KARST, E. New quadratic forms with high density of primes. Elem. d. Math., 28,1973,116-118. 1974 GOLOMB, S.W. A direct interpretation of Gandhi's formula. Amer. Math. Monthly, 81,1974,752-754. 1974 HENDY, M.D. Prime quadratics associated with complex quadratic fields of class number two. Proc. Amer. Math. Soc., 43, 1974,253-260. 1974 MONTGOMERY, H.L. and WEINBERGER, P.J. Notes on small class numbers. Acta Arithm., 24, 1974, 529-542. 1974 NARKIEWICZ, W. Elementary and Analytic Theory of Algebraic Numbers. Polish Sci. Publ., Warsaw, 1974. Chapter 3 465

1974 SZEKERES, G. On the number of divisors of x 2 + x + A. J. Nb. Th., 6, 1974, 434-442. 1975 BAKER, A. Transcendental Number Theory. Cambridge Univ. Press, Cambridge, 1975. 1975 ERNV ALL, R. A formula for the least prime greater than a given integer. Elem. d. Math., 30, 1975, 13-14. 1975 JONES, J.P. Diophantine representation of the Fibo­ nacci numbers. Fibonacci Quart., 13, 1975,84-88. 1975 MATIJASEVIC, YD. V. Reduction of an arbitrary diophantine equation to one in 13 unknowns. Acta Arithm., 27, 1975,521-553. 1975 STARK, H.M. On complex quadratic fields with class-number two. Math. Comp., 29,1975,289-302. 1976 CHOWLA, S. and FRIEDLANDER, J.B. Class num­ bers and quadratic residues. Glasgow Math. J., 17, 1976, 47-52. 1976 GOLDFELD, D. The class number of quadratic fields and conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa, Ser. 4, 3, 1976, 623-663. 1976 JONES, J.P., SATO, D., WAD A, H. and WIENS, D. Diophantine representation of the set of prime numbers. Amer. Math. Monthly, 83, 1976,449-464. 1977 MATIJASEVIC, YD. V. Primes are nonnegative values of a polynomial in 10 variables. Zapiski Sem. Lenin­ grad Mat. Inst. Steklov, 68, 1977, 62-82. English translation by L. Guy and J.P. Jones, J. Soviet Math., 15, 1981, 33-44. 1977 MATIJASEVIC, YD. V. Some purely mathematical results inspired by mathematical logic, in Logic, Founda­ tions of Mathematics and Computability Theory (Proc. 5th Intern. Congress Logic, Methodology and Philosophy of Science, Univ. Western Ont., London, Ont., 1975), 121- 127. Reidel, Dordrecht, 1977. 1979 JONES, J.P. Diophantine representation of Mersenne and Fermat primes. Acta Arithm., 35,. 1979, 209- 221. 1980 FELTHI, C. Non-nullite des fonctions zeta des corps quadratiques reels pour 0 < s < 1. c.R. Acad. Sci. Paris, Ser. A, 291, 1980,623-625. 466 Bibliography

1980 KUTSUNA, M. On a criterion for the class number of a quadratic number field to be one. Nagoya Math. J., 79, 1980, 123-129. 1981 AYOUB, R.G. and CHOWLA, S. On Euler's polyno­ mial. J. Nb. Th., 13, 1981,443-445. 1981 HATCHER, W.S. and HODGSON, B.R. Complexity bounds on proof. J. Symb. Logic, 46, 1981,255-258. 1982 JONES, J.P. Universal diophantine equation. J. Symb. Logic, 47, 1982, 549-571. 1983 GROSS, B. and ZAGIER, D. Points de Heegner et derivees de fonctions L. c.R. Acad. Sci. Paris, 297, 1983, 85-87. 1984 OESTERLE, J. Nombres de classes des corps quadra­ tiques imaginaires. Seminaire Bourbaki, expose No. 631, 1983/4, Paris. Asterisque, 121/2, 1985,213-224. 1985 GOLDFELD, D. Gauss' class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc., 13, 1985,23-37. 1986 SASAKI, R. On a lower bound for the class number of an imaginary quadratic field. Proc. Japan Acad., Ser. A, 62, 1986, 37-39. 1988 DUBNER, H. Prime triplets in arithmetic progressions starting with 3. J. Recr. Math., 20,1988,211-213. 1988 GARDNER, M. Primes in arithmetic progression, in The Mathematical Sciences Calendar for 1988, Rome Press, Raleigh, N.C., 1987. 1988 MOLLIN, R.A. Necessary and sufficient conditions for the class number of a real quadratic field to be 1, and a conjecture of Chowla. Proc. Amer. Math. Soc., 102, 1988, 17-21. 1988 MOLLIN, R.A. and WILLIAMS, H.C. A conjecture of S. Chowla via the generalized Riemann hypothesis. Proc. Amer. Math. Soc., 102, 1988, 794-796. 1988 MOLLIN, R.A. and WILLIAMS, H.C. On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J., 112,1988,143-151. 1988 RIBENBOIM, P. Euler's famous prime generating polynomial and the class number of imaginary quadratic fields. L'Enseign. Math., 34, 1988, 23-42. Chapter 4 467

1989 COX, D.A. Primes of the Form x 2 + ny2. Wiley­ Interscience, New York, 1989. 1989 FLATH, D.E. Introduction to Number Theory. Wiley­ Interscience, New York, 1989. 1989 GOETGHELUCK, P. On cubic polynomials giving many primes. Elem. d. Math., 44, 1989, 70-73. 1990 LOUBOUTIN, S. Minorations (so us l'hypothese de Riemann generalisee) des nombres de classes des corps quadratiques imaginaires. c.R. Acad. Sci. Paris, Ser. I, 310, 1990, 795-800. 1990 LOUBOUTIN, S. Prime producing quadratic polyno­ mials and class numbers of real quadratic fields. Can. J. Math., 42, 1990,315-341. 1991 LOUBOUTIN, S. Extensions du theoreme de Frobenius­ Rabinowitsch. c.R. Acad. Sci. Paris, Ser. I, 310,1991,711-714. 1991 RIBENBOIM, P. Gauss and the class number problem. Symp. Gaussiana, 1, 1991, 1-63. 1993 MOLLIN, R.A. Orders in quadratic fields, I. Proc. Japan Acad., Ser. A, 69,1993,45-48. 1994 MOLLIN, R.A. Quadratic polynomials producing con­ secutive distinct primes and class groups of complex quadratic fields. Preprint (1994), 52 pages, to appear in Acta Arithm. 1994 MOLLIN, R.A. Quadratic prime producing polyno­ mials and class groups of real quadratic tields. Preprint (1994),46 pages. 1995 BOSTON, N. and GREENWOOD, M.L Quadratics representing primes. Amer. Math. Monthly, 102, 1995, 595-599. 1995 GOSTIN, G. New factors of Fermat numbers. To appear in Math. Compo (1995). 1995 MOLLIN, R.A. Quadratics. e.R.e. Press, 1995.

CHAPTER 4 1815 CAUCHY, A.L. Demonstration generale du theoreme de Fermat sur les nombres polygones. Bull. Soc. Philomatique, 1815, 196-197. Reprinted in Oeuvres, (2), Vol. 2, 202-204. Gauthier-Villars, Paris, 1903. 468 Bibliography

1837 DIRICHLET, G.L. Beweis des Satzes, dass jede un be­ grenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthalt. Abh. d. Konigl. Akad. d. Wiss., 1837, 45-81. Reprinted in Werke, Vol. I, 315-350. G. Reimer, Berlin, 1889. 1878 PROTH, F. Sur la serie des nombres premiers. N OUD. Corresp. Math., 4,1878,236-240. 1885 MEISSEL, KD.F. Berechnung der Menge von Prim­ zahlen, welche innerhalb der ersten Milliarde naturlicher Zahlen vorkommen. Math. Ann., 25,1885,251-257. 1892 SYLVESTER, J.J. On arithmetical series. Messenger of Math., 21, 1892, 1-19 and 87-120. Reprinted in Gesammelte Abhandlungen, Vol. III, 573-587. Springer­ Verlag, New York, 1968. 1895 WENDT, E. Elementarer Beweis des Satzes dass in jeder unbegrenzter arithmetischen Progression my + 1 unendlich viele Primzahlen vorkommen. Journal f d. reine u. angew. Math., 115, 1895,85-88. 1901 VON KOCH, H. Sur la distribution des nombres premiers. Acta Math., 24,1901,159-182. 1901 WOLFSKEHL, P. Ueber eine Aufgabe der elementaren Arithmetik. Math. Ann., 54, 190!, 503-504. 1902 TORELLI, G. Sulla totalita dei numeri primi fino a un limite assegnato. Atti d. Reale Acad. d. Sci. Fis. e Mat. di Napoli, (2), 11, 1902, 1-222. 1903 LANDAU, K Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Math. Ann., 56, 1903,645-670. 1909 HILBERT, D. Beweis fur die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem). Math. Ann., 67, 1909, 281-300. Reprinted in Gesammelte Abhandlungen, Vol. I, 510-527, 2nd edition, Chelsea, Bronx, N.Y., 1965. 1909 LANDAU, K Handbuch der Lehre von der Verteilung der Primzahlen. Teubner, Leipzig, 1909. Reprinted by Chelsea, Bronx, N.Y., 1974. 1909 LEHMER, D.N. List of Prime Numbers from 1 to 10,006,721. Reprinted by Hafner, New York, 1956. Chapter 4 469

1909 LEHMER, D.N. Factor Table for the First Ten Millions Containing the Smallest Factor of Every Number not Divisible by 2, 3, 5 or 7, Between the Limits ° and 10,017,000. Reprinted by Hafner, New York, 1956. 1909 WIEFERICH, A. Beweis des Satzes, dass sich eine jede ganze Zahl als Summe von hochstens neun positiven Kuben darstellen lasst. Math. Ann., 66, 1909, 95~ 101. 1912 SCHUR, I. Uber die Existenz unendlich vieler Prim­ zahlen in einiger speziellen arithmetischen Progressionen. Sitzungsber. Berliner Math. Ges. 11, 1912,40--50. 1912 STRIDSBERG, E. Sur la demonstration de M. Hilbert du theoreme de Waring. Math. Ann., 72, 1912, 145~ 152. 1913 COBLYN Sur les couples de nombres premiers. Soc. Math. France, c.R., 1913, 55~ 56. 1914 LITTLEWOOD, J.E. Sur la distribution des nombres premiers. c.R. Acad. Sci. Paris, 158, 1914, 1869~ 1872. 1916 WEYL, H. Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77, 1916, 313~352. 1919 BRUN, V. Le crible d'Eratosthene et Ie theoreme de Goldbach. c.R. Acad. Sci. Paris, 168, 1919, 544~546. 1919 BRUN, V. La serie i + ~ + /1 + /3 + 1~ + /9 + 219+ 1 1 1 1 1 'I d' . TI + 41 + 43 + 59 + 6T + . .. ou es enormnateurs sont "nombres premiers jumeaux" est convergente ou finie. Bull. Sci. Math., (2), 43, 1919, 100~ 104 and 124~ 128. 1919 RAMANUJAN, S. A proof of Bertrand's postulate. J. Indian Math. Soc., 11, 1919, 181~182. Reprinted in Collected Papers (edited by G.H. Hardy, P.v. Seshu Aiyar, and B.M. Wilson), 208~209. Cambridge University Press, Cambridge, 1927. Reprinted by Chelsea, Bronx, N.Y., 1962. 1920 BRUN, V. Le crible d'Erathostene et la theoreme de Goldbach. Videnskapsselskapets Skrifter Kristiania, M at.­ nat. Kl. 1920, No.3, 36 pages. 1920 HARDY, G.H. and LITTLEWOOD, J.E. A new solution of Waring's problem. Quart. J. Math., 1920, 48, 272~293. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 382~393. Clarendon Press, Oxford, 1966. 1920 HARDY, G.H. and LITTLEWOOD, J.E.. Some prob­ lems of "Partitio Numerorum", I: A new solution of 470 Bibliography

Waring's Problem. Nachr. Konigl. Ges. d. »iss. zu Gottingen, 1920, 33-54. Reprinted in Collected Papers of G.H. Hardy, Vol. 1,405-426. Clarendon Press, Oxford, 1966. 1921 HARDY, G.H. and LITTLEWOOD, J.E. Some prob­ lems of "Partitio Numerorum", II: Proof that every number is the sum of at most 21 biquadrates. Math. Zeits., 9, 1921, 14-27. Reprinted in Collected Papers of G.H. Hardy, Vol. 1,427-440. Clarendon Press, Oxford, 1966. 1921 KAMKE, E. Verallgemeinerung des Waring-Hilbert­ schen Satzes. Math. Ann., 83, 1921,85-112. 1922 HARDY, G.H. and LITTLEWOOD, J.E. Some prob­ lems of "Partitio Numerorum", IV: The singular series in Waring's Problem and the value of the number G(k). Math. Zeits., 12,1922,161-188. Reprinted in Collected Papers of G.H. Hardy, Vol. 1,441-468. Clarendon Press, Oxford, 1966. 1923 HARDY, G.H. and LITTLEWOOD, J.E. Some prob­ lems of "Partitio Numerorum", III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1-70. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 561-630. Clarendon Press, Oxford, 1966. 1923 NAGELL, T. Zahlentheoretische Notizen I, Ein Beitrag zur Theorie der h6heren Kongruenzen. Vidensk. Skrifter, ser. I, Math. Nat. Kl., No. 13, Kristiania, 1923,3-6. 1924 RADEMACHER, H. Beitdige zur Viggo Brunschen Methode in der Zahlentheorie. Abh. Math. Sem. U niv. Hamburg, 3, 1924, 12-30. Reprinted in Collected Papers (edited by E. Grosswald), Vol. I, 280-288. M.I.T. Press, Cambridge, 1974. 1925 HARDY, G.H. and LITTLEWOOD, J.E. Some prob­ lems of "Partitio Numerorum," VI. Further researches in Waring's problem. Math. Z., 23, 1925, 1-37. Reprinted in Collected Papers of G.H. Hardy. Vol. I, 469-505. Clarendon Press, Oxford, 1966. 1928 SCHOENBERG, I.J. Uber die asymptotische Vertei­ lung reeller Zahlen (mod 1). Math. Zeits., 28,1928,171-199. 1928 WINTNER, A. Uber den Konvergenzbegriff der mathe­ matischen Statistik. Math. Zeits., 28, 1928,476-480. 1930 ERDOS, P. Beweis eines Satzes von Tschebychef. Acta Sci. Math. Szeged, 5,1930,194-198. Chapter 4 471

1930 HOHEISEL, G. Primzahlprobleme in der Analysis. Sitzungsberichte Berliner Akad. d. »iss., 1930,580-588. 1930 SCHNIRELMAN, G. Uber additive Eigenschaften von Zahlen. Ann. Inst. Polytechn. Novocerkask, 14, 1930, 3-28 and Math. Ann., 107, 1933,649-690. 1930 TITCHMARSH, E.C. A divisor problem. Rend. Cir. Mat. Palermo, 54, 1930,414-429. 1931 WESTZYNTHIUS, E. Uber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Comm. Phys. Math. Helsingfors, (5), 25, 1931, 1-37. 1932 INGHAM, A.E. The Distribution of Prime Numbers, Cambridge Tracts in Math. 30, Cambridge University Press, 1932 (second edition, 1990, with a Foreword by R.c. Vaughan). 1932 LANDAU, E. Uber den Wienerschen neuen Weg zum Primidealsatz. Sitzungsber. Berliner Akad. d. Wiss., 1932, 514-521. 1933 SKEWES, S. On the difference n(x) - Ii (x). J. London Math. Soc., 8, 1933,277-283. 1934 CHOWLA, S. On the least prime in an arithmetic progression. J. Indian Math. Soc., (2),1, 1934, 1-3. 1934 ISHIKA W A, H. Uber die Verteilung der Primzahlen. Sci. Rep. Tokyo Bunrika Daigaku, A, 2, 1934,27-40. 1935 ERDOS, P. On the density of some sequences of numbers, I. J. London Math. Soc., 10,1935,120-125. 1935 ERDOS, P. On the difference of consecutive primes. Quart. J. Pure and Appl. Math., Oxford, 6, 1935, 124- 128. 1935 ERDOS, P. On the normal number of prime factors of p - 1 and some related problems concerning Euler's tP-function. Quart. J. Pure and Appl. Math., Oxford, 6, 1935, 205-213. 1935 PILLAI, S.S. On Waring's problem, I. Annamalai Vniv. J., 5, 1935, 145-166. 1935 VINOGRADOV,I.M. On Waring's problem. Annals of Math., 36, 1935,395-405. 1936 CUDAKOV, N.G. On zeros of the function '(s) (in Russian). c.R. Acad. Sci. V.R.S.S., 1 (x), 1936, 201- 204. 472 Bibliography

1936 DICKSON, L.E. Solution of Waring's problem. Amer. J. Math., 58, 1936, 530-535. Reprinted in The Collected Mathematical Papers (edited by A.A. Albert), Vol. III, 290-295. Chelsea, Bronx, N.Y., 1975. 1936 HEILBRONN, H. Uber das Waringsche Problem. Acta Arithm., 1, 1936,212-221. 1936 PILLAI, S.S. On Waring's problem, III, IV, Annamalai Univ. J., 6, 1936, 50-64. 1937 CRAMER, H. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithm., 2, 1937, 23-46. 1937 ERDOS, P. On the density of some sequences of numbers, II. J. London Math. Soc., 12,1937,7-11. 1937 INGHAM, A.E. On the difference between consecutive primes. Quart. J. Pure and Appl. Math., Oxford, Ser. 2, 8, 1937,255-266. 1937 LANDAU, E. Uber einige neuere Fortschritte der additiven Zahlentheorie. Cambridge University Press, Cambridge, 1937. Reprinted by Stechert-Hafner, New York, 1964. 1937 TURAN, P. Uber die Primzahlen der arithmetischen Progressionen. Acta Sci. Math. Szeged, 8, 1937,226-235. 1937 VAN DER CORPUT, J.G. Sur l'hypothese de Gold­ bach pour presque tous les nombres pairs. Acta Arithm., 2, 1937,266-290. 1937 VINOGRADOV, I.M. Representation of an odd num­ ber as the sum of three primes (in Russian). Dokl. Akad, Nauk SSSR, 15,1937,169-172. 1937 VINOGRADOV, I.M. Some new problems in the theory of primes (in Russian). Dokl. Akad. Nauk SSSR, 16, 1937, 131-132. 1937 VINOGRADOV, I.M. Some theorems concerning the theory of primes (in Russian). Mat. Sbornik, N.S., 2, (44), 1937, 179-195. 1938 ERDOS, P. On the density of some sequences of numbers, III. J. London Math. Soc., 13, 1938, 119-127. 1938 ESTERMANN, T. Proof that almost all even positive integers are sums of two primes. Proc. London Math. Soc., 44,1938,307-314. Chapter 4 473

1938 HARDY, G.H. and HEILBRONN, H. Edmund Lan­ dau. J. London Math. Soc., 13,1938,302-310. Reprinted in Collected Papers of G.H. Hardy, Vol. VII, 762-770. Clarendon Press, Oxford, 1979. 1938 POULET, P. Table des nombres composes verifiant Ie theoreme de Fermat pour Ie module 2, jusqu'a 100.000.000. Sphinx, 8, 1938, 52-52. Corr,ections: Math. Camp., 25, 1971,944-945 and 26, 1972, 814. 1938 RANKIN, R.A. The difference between consecutive prime numbers. J. London Math. Soc., 13, 1938,242-247. 1938 ROSSER, J.B. The nth prime is greater than n log n. Proc. London Math. Soc., 45, 1938,21-44. 1938 CUDAKOV, N.G. On the density of the set of even integers which are not representable as a sum of two odd primes (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 1, 1938, 25-40. 1938 VINOGRADOV, I.M. Some general theorems about primes (in Russian). Trav. Inst. Math. Tbilissi, 3, 1938, 35-67. 1939 DAVENPORT, H. On Waring's probkm for cubes. Acta Math., 71, 1939, 123-143. 1939 DAVENPORT, H. On Waring's problem for fourth powers. Annals of Math., 40,1939,731-747. 1939 DICKSON, L.E. All integers, except 23 and 239, are sums of 8 cubes. Bull. Amer. Math. Soc., 45, 1939,588-591. Reprinted in The Collected Mathematical Papers (edited by A.A. Albert), Vol. V, 66-69. Chelsea, Bronx, N.Y., 1975. 1939 VAN DER CORPUT, J.G. Uber Summen von Primzahlen und Primzahlquadraten. Math. Ann., 116, 1939,1-50. 1940 ERDOS, P. The difference of consecutive primes. Duke Math. J., 6,1940,438-441. 1940 INGHAM, A.E. On the estimation of N(11, T). Quart. J. Math., Oxford, 11, 1940,291-292. 1940 PILLAI, S.S. On Waring's problem: g(6) = 73. Proc. Indian Acad. Sci., A, 12, 1940, 30-40. 1941 NARASIMHAMURTI, V. On Waring's problem for 8th, 9th and 10th powers. J. Indian Math. Soc., 5, 1941, 122. 474 Bibliography

1942 DAVENPORT, H. On Waring's problem for fifth and sixth powers. Amer. J. Math., 64,1942, 199-207. 1942 MANN, H.B. A proof of the fundamental theorem of sums of sets of positive integers. Annals of Math., 43, 1942, 523-527. 1942 RUBUGUNDAY, R. On g(k) in Waring's problem. J. Indian Math. Soc., N.S., 6,1942,192-198. 1943 ARTIN, E. and SCHERK, P. On the sums of two sets of integers. Annals of Math., 44, 1943, 138-142. Reprinted in Collected Papers (edited by S. Lang and J.T. Tate), 346-350. Addison-Wesley, Reading, Mass., 1965. 1943 LINNIK, Yu.V. An elementary solution of a problem of Waring by Schnirelman's method. Mat. Sbornik, 12, (54), 1943,225-230. 1943 SELBERG, A. On the normal density of primes in small intervals and the difference between consecutive primes. Arch. Math. Naturid., 47,1943, (6),87-105. 1944 CHOWLA, S. There exists an infinity of 3-combinations of primes in A. P. Proc. Lahore Phil. Soc., 6, 1944, 15-16. 1944 LINNIK, Yu.V. On the least prime in an arithmetic progression I. The basic theorem (in Russian). Mat. Sbornik, 15 (57), 1944, 139-178. 1944 NIVEN, I. An unsolved case of the Waring problem. Amer. J. Math., 66,1944,137-143. 1945 ERDOS, R. Some remarks on Euler's ¢-function and some related problems. Bull. Amer. Math. Soc., 51, 1945, 540-544. 1946 BRAUER, A. On the exact number of primes below a given limit. Amer. Math. Monthly, 9,1946,521-523. 1946 JOFFE, S.A. Review of Kulik's "Magnus Canon Divisorum ..." Math. Comp., 2,1946/7,139-140. 1947 KHINCHIN, A.Ya. Three Pearls of Number Theory. Original Russian edition in OGIZ, Moscow, 1947. Translation into English published by Graylock Press, Baltimore, 1952. 1947 RENYI, A. On the representation of even numbers as the sum of a prime and an almost prime. Dokl. Akad. N auk SSSR, 56, 1947,455-458. Chapter 4 475

1947 VINOGRADOV, I.M. The method of trigonometrical sums in the theory of numbers (in Russian). Trav. Inst. Math. Steklov, 23, 1947, 109 pages. 1948 ERDOS, P. and TURiN, P. On some new questions on the distribution of prime numbers. Bull. Amer. Math. Soc., 54, 1948,371-378. 1948 VINOGRADOV, I.M. Uber die Abschatzung trigono­ metrischer Summen mit Primzahlen. Izv. Akad. Nauk SSSR, Ser. Mat., 12, 1948,225-248. 1949 CLEMENT, P.A. Congruences for sets of primes. Amer. Math. Monthly, 56, 1949,23-25. 1949 ERDOS, P. Problems and results on the difference of consecutive primes. Publ. Math. Debrecen, 1, 1949, 33- 37. 1949 ERDOS, P. On a new method in elementary number the­ ory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A., 35, 1949, 374- 384. 1949 ERDOS, P. On the converse of Fermat's theorem. Amer. Math. Monthly, 56, 1949,623-624. 1949 MOSER, L. A theorem on the distribution of primes. Amer. Math. Monthly, 56, 1949,624-625. 1949 RICHERT, H.E. Uber Zerfallungen in ungleiche Prim­ zahlen. Math. Zeits., 52,1949,342-343. 1949 SELBERG, A. An elementary proof of the prime number theorem. Annals oj Math., 50, 1949, 305-313. 1949 SELBERG, A. An elementary proof of Dirichlet's theo­ rem about primes in an arithmetic progression. Annals oj Math., 50, 1949,297-304. 1949 SELBERG, A. An elementary proof of the prime number theorem for arithmetic progressions. Can. J. Math., 2, 1950,66-78. 1950 ERDOS, P. On some application of Brun's method. Acta Sci. Math. Szeged, 13, 1950,57-63. 1950 ERDOS, P. On almost primes. Amer. Math. Monthly, 57, 1950, 404-407. 1950 HASSE, H. Vorlesungen iiber Zahlentheorie. Springer­ Verlag, Berlin, 1950. 476 Bibliography

1950 SELBERG, A. The general sieve method and its place in prime number theory. Proc. Int. Congr. Math., Cambridge, 1950. 1950 SHAPIRO, H.N. On a theorem of Selberg and generalization. Annals of Math., (2), 51, 1950, 485~497. 1951 NAGELL, T. Introduction to Number Theory. Almqvist and Wiksell, Stockholm, 1964. Reprinted by Chelsea, Bronx, N.Y., 1964. 1951 TITCHMARSH, E.c. The Theory of the Riemann Zeta-Function. Clarendon Press, Oxford, 1951 (second edition, 1986). 1951 WATSON, G.L. A proof of the seven cube theorem. J. London Math. Soc., 26, 1951, 153~ 156. 1952 NAGURA, J. On the interval containing at least one prime number. Proc. Japan Acad., 28, 1952, 177~ 181. 1952 WRIGHT, E.M. The elementary proof of the prime number theorem. Proc. Royal Soc. Edinburgh A63, 1952, 257~267. 1953 PYATETSKII-SHAPIRO, 1.1. On the distribution of prime numbers in sequence of the form [f(n)] (in Russian). Mat. Sbornik, N.S. 33 (75), 1953, 559~566. 1953 RIEGER, G.J. Zur Hilbertschen Lasung der Waring­ schen Problems: Abschatzung von g(n). Arch. d. Math., 4, 1953, 275~281. 1954 BREUSCH, R. Another proof of the prime number theorem. Duke Math. J., 21, 1954, 49~53. 1954 RIEGER, G.J. Zu Linniks Lasung des Waringsches Problems: Abschatzung von g(n). Math. Zeits., 60, 1954, 213~234. 1955 F JELLSTEDT, L. Bemerkungen tiber gleichzeitige Lasbarkeit von Kongruenzen. Arkiv Mat., 3, 1955, 193~ 198. 1955 RICCI, G. Recherches sur l'allure de la suite {(Pn+1 - Pn)/logPn}. Coli. Th. Nombres Bruxelles 1955, 93~106. G. Thone, Liege, 1956. 1955 SKEWES, S. On the difference n(x) - h(x), II. Proc. London Math. Soc., 5, 1955, 48~ 70. 1956 AMITSUR, S.A. On arithmetic functions. J. Anal. Math., 5, 1956/7, 273~314. Chapter 4 477

1956 BORODZKIN, K.G. On the problem of 1.M. Vino­ gardov's constant (in Russian). Proc. Third Math. Con­ gress, Moscow, 1, (1956). 1956 ERDOS, P. On pseudo-primes and Carmichael num­ bers. Publ. Math. Debrecen, 4, 1956,201-206. 1956 NIVEN, I. Irrational Numbers. Carus Math. Mono­ graphs, No. 11. Math. Assoc. of America, Washington, 1956. 1956 OSTMANN, H.H. Additive Zahlentheorie (2 volumes). Springer-Verlag, Berlin, 1956 (2nd edition 1969). 1957 FELLER, W. An Introduction to Probability Theory and its Applications, Vol. 1. 1. Wiley and Sons, New York, 1957. 1957 HUA, L.K. Additive Theory of Prime Numbers. Inst. Math. Chinese Acad. Sciences, Peking, 1957. Translated into English by H.H. Ng. Amer. Math. Soc., Providence, 1965. 1957 LEECH, J. Note on the distribution of prime numbers. J. London Math. Soc., 32, 1957,56-58. 1957 MAHLER, K. On the fractional parts of powers of real numbers. Matematika, 4, 1957, 122-124. 1958 BAKER, c.L. and GRUENBERGER, F.J. Primes in the Thousandth Million. The Rand Corp., Santa Monica, 1958. 1958 CHEN, J.R. On Waring's problem for nth powers. Acta Math. Sinica, 8, 1958, 253-257; Chinese "Math. Acta, 8, 1966/7,849-853. 1958 KOROBOV, N.M. Estimates of trigonometric sums and their applications (in Russian). U sp. N( at. N auk, 13, 1958, 185-192. 1958 SCHINZEL, A. and SIERPINSKI, W. Sur certaines hypotheses concernant les nombres premiers. Acta Arithm., 4, 1958, 185-208; Erratum, 5, 1959,259. 1958 VINOGRADOV, J.M. A new estimate for ((1 + it). Izv. Akad. Nauk S.S.S.R., Ser. Mat., 22,1958,161-164. 1959 BAKER, c.L. and GRUENBERGER, F.J. The First Six Million Prime Numbers. Microcard Found., Madison, 1959. 1959 HUA, L.K. Abschatzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie. Enzykl. d. Math. Wiss., 12, Heft 13, t.1. Teubner, Leipzig, 1959. 478 Bibliography

1959 KAC, M. Statistical Independence in Probability, Analysis and Number Theory. Carus Math. Monographs, No. 12, Math. Assoc. of America, Washington, 1959. 1959 KILL GROVE, R.B. and RALSTON, K.E. On a conjecture concerning the primes. Math. Camp., 13, 1959, 121-122. 1959 LEHMER, D.H. On the exact number of primes less than a given limit. Illinois J. Math., 3, 1959, 381-388. Reprinted in Selected Papers (edited by D. McCarthy), Vol. III, 1104-1111. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981. 1959 SCHINZEL, A. Demonstration d'une consequence de l'hypothese de Goldbach. Compositio Math., 14, 1959, 74-76. 1959 SCHINZEL, A. Sur une consequence de l'hypothese de Goldbach. Izvestija Mat. Inst., Bulgarian Acad. Sci., 4, 1959, 35-38. 1959 SHANKS, D. Quadratic residues and the distribution of primes. Math. Comp., 13,1959,272-284. 1959 VINOGRADOV, I.M. On an upper bound for G(n) (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 23, 1959, 637-642. 1960 JACOBSTHAL, E. Uber Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, I, II, III. N orske Videnskabsselskab F orhdl., 33, 1960, 117-139. 1960 NEWMAN, D.J. A simplified proof of Waring's conjecture. Michigan Math. J., 7, 1960,291-295. 1961 JURKAT, W.B. Eine Bemerkung zur Vermutung von Mertens. Machr. der Osterr. Math. Ges., Sondernummer Ber. V. Osterr. Math.-Kongress, Vienna, 1961, 11. 1961 PRACHAR, K. Uber die kleinste Primzahl einer arithmetischen Reihe. Journal f d. reine u. angew. Math., 206, 1961,3-4. 1961 ROTKIEWICZ, A. Demonstration arithmetique d'exis­ tence d'une infinite de nombres premiers de la forme nk + 1. L'Enseign. Math., (2), 7, 1962,277-280. 1961 WRENCH, J.W. Evaluation of Artin's constant and the twin-prime constant. Math. Camp., 15, 1961,396-398. Chapter 4 479

1962 ERDOS, P. On the integers relatively prime to n and on a number theoretic function considered by lacobsthal. Math. Scand., to, 1962, 163-170. 1962 ROSSER, J.B. and SCHOENFELD, L. Approximate formulas for some functions of prime numbers. Illinois J. Math., 6, 1962,64-94. 1962 SCHINZEL, A. Remark on a paper of K. Prachar, "Uber die kleinste Primzahl einer arithmetischen Reihe". Journal! d. reine u. angew. Math., 2tO, 1962, 121-122. 1962 SEGAL, S. On n(x + y) ::;; n(x) + n(y). Trans. Amer. Math. Soc., 104, 1962,523-527. 1963 AYOUB, R.G. An Introduction to the Theory of Numbers. Amer. Math. Soc., Providence, R.I., 1963. 1963 ESTERMANN, T. Note on a paper of A. Rotkiewicz. Acta Arithm., 8, 1963, 465-467. 1963 KANOLD, H.J. Elementare Betrachtungen zur Prim­ zahltheorie. Arch. Math., 14, 1963, 147-151. 1963 NEUBAUER, G. Eine empirische Unte:rsuchung zur Mertenssche Funktion. N umer. Math., 5, 1963, 1-13. 1963 RANKIN, R.A. The difference between consecutive prime numbers, V. Proc. Edinburgh Math. Soc., (2), 13, 1963, 331-332. 1963 ROTKIEWICZ, A. Sur les nombres ps,;!udo-premiers de la forme ax + b. c.R. Acad. Sci. Paris, 257, 1963, 2601-2604. 1963 SCHONHAGE, E. Eine Bemerkung zur Konstruktion grosser Primzahllucken. Archiv d. Math., 14, 1963,29-30. 1963 W ALFISZ, A.Z. Weylsche Exponentialsummen in der neueren Zahlentheorie. VEB Deutscher Verlag d. Wiss., Berlin, 1963. 1964 CHEN, J.R. Waring's problem for g(5) = 37. Sci. Sinica, 13, 1964, 1547-1568. Reprinted in Chinese Jyf athematics, 6, 1965, 105-127. 1964/1965 KANOLD, H.J. Uber Primzahlen in arithmeti­ schen Folgen. Math. Ann., 156, 1964, 393-395; 157, 1965, 358-362. 1964 KAPFERER, H. Verifizierung des symmetrischen Teils der Fermatschen Vermutung fur unendlich viele paarweise 480 Bibliography

teilerfremde Exponenten E. Journal f d. reine u. angew. Math., 214/5, 1964,360-372. 1964 ROHRBACH, H. and WEIS, J. Zum finiten Fall des Bertrandschen Postulats. Journal f d. reine u. angew. Math., 214/5,1964,432-440. 1964 SHEN, M.K. On checking the Goldbach conjecture. Nordisk Tidskr., 4, 1964,243-245. 1964 SIERPINSKI, W. Elementary Theory of Numbers. Polish Scientific PubIs, Warszawa, 1964. (2nd edition, revised A. Schinzel, North Holland, Amsterdam, 1987). 1964 STEMMLER, R.M. The ideal Waring theorem for exponents 401-200,000. Math. Comp., 18, 1964, 144- 146. 1965 BATEMAN, P.T. and LOW, M.E. Prime numbers in arithmetic progression with difference 24. Amer. Math. Monthly, 72, 1965, 139-143. 1965 GELFOND, A.O. and LINNIK, Yu.V. Elementary Methods in Analytic Number Theory. Translated by A. Feinstein, revised and edited by L.l. Mordell. Rand McNally, Chicago, 1965. 1965 PAN, C.D. On the least prime in an arithmetic progression. Sci. Record (N.S.) 1, 1957,311-313. 1965 ROTKIEWICZ, A. Les intervalles con tenant les nom­ bres pseudo premiers. Rend. Circ. Mat. Palermo (2), 14, 1965,278-280. 1965 STEIN, M.L. and STEIN, P.R. New experimental results on the Goldbach conjecture. Math. Mag., 38, 1965, 72-80. 1965 STEIN, M.L. and STEIN, P.R. Experimental results on additive 2-bases. Math. Comp., 19, 1965,427-434. 1965 WANG Y., HSIEH, S.K., and YU, K. Two results on the distribution of prime numbers (in Chinese). J. China Univ. Sci. Techn. 1 (1965), 32-38. 1966 BEILER, A.H. Recreations in the Theory of Numbers (The Queen of Mathematics Entertains). Dover, New York, 1966. 1966 BOMBIERI, E. and DAVENPORT, H. Small differ­ ences between prime numbers. Proc. Roy. Soc., A, 293, 1966, 1-18. Chapter 4 481

1966 CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao, 17, 1966, 385-386. 1966 LEHMAN, R.S. On the difference n(x) - li(x). Acta Arithm., 11, 1966,397-410. 1967 JARDEN, D. Existence of arbitrarily long sequences of consecutive numbers in arithmetic progressions divisible by arbitrarily many different primes. Fibonacci Quart., 5, 1967,287. 1967 JONES, M.F., LAL, M., and BLUNDON, W.J. Statis­ tics on certain large primes. Math. Comp., 21, 1967, 103-107. 1967 KOLESNIK, G.A. The distribution of primes in

sequences of the form [n C ] (in Russian). Mat. Zametki, 2, 1967,117-128. 1967 LANDER, L.J. and PARKIN, T.R. Consecutive primes in arithmetic progression. Math. Comp., 21, 1967,489. 1967 ROTKIEWICZ, A. On the pseudo-primes of the form ax + b. Proc. Cambridge Phil. Soc., 63, 1967,389-392. 1967 SZYMICZEK, K. On pseudo-primes which are prod­ ucts of distinct primes. Amer. Math. Monthly, 74, 1967, 35-37. 1968 HALBERSTAM, H. and ROTKIEWICZ, A. A gap theorem for pseudo primes in arithmetic progressions. Acta Arithm., 13, 1968, 395-404. 1969 GROLZ, W. Primteiler von Polynomen. Math. Ann., 181, 1969, 134-136. 1969 MONTGOMERY, H.L. Zeros of L-func:tions. Invent. Math., 8, 1969,346-354. 1969 NAGELL, T. Sur les diviseurs premiers des polynomes. Acta Arithm., 15, 1969,235-244. 1969 RICHERT, H.E. Selberg's sieve with weights. Mathe­ matika, 16, 1969, 1-22. 1969 ROSSER, J.B., YOHE, J.M. and SCHOENFELD, L. Rigorous computation of the zeros of the Riemann zeta-function (with discussion). Inform. Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), Vol. I, 70-76. North-Holland, Amsterdam, 1969. 482 Bibliography

1969 SCHOENFELD, L. An improved estimate for the summatory function of the Mobius function. Acta Arithm., 15, 1969,221-233. 1969 WOJCYK, J. A refinement of a theorem of Schur on primes in arithmetic progressions, III. Acta Arithm., 15, 1969, 193-197. 1970 DRESSLER, R.E. A density which counts multiplicity. Pacific J. Math., 34, 1970, 371-378. 1970 HORNFECK, B. Primteiler von Polynomen. Journal f d. reine u. angew. Math., 243, 1970, 120. 1970 MOTOHASHI, Y. A note on the least prime in an arithmetic progression with a prime difference. Acta Arithm., 17,1970,283-285. 1970 SERRE, J.P. Cours d'Arithmhique. Presses Univ. France, Paris, 1970. English translation published by Springer-Verlag, New York, 1973. 1971 ELLIOTT, P.D.T.A. and HALBERSTAM, H. The least prime in arithmetic progression. Studies in Pure M athemat­ ics (edited by R Rado), 59-61. Academic Press, London, 1971. 1971 ELLISON, W.J. Waring's problem. Amer. Math. Monthly, 78,1971,10-36. 1971 GERST, I. and BRILLHART, J. On prime divisors of polynomials. Amer. Math. Monthly, 78,1971,250-266. 1971 MONTGOMERY, H.L. Topics in Multiplicative Num­ ber Theory. Lecture Notes in Math., #227. Springer­ Verlag, New York, 1971. 1971 SERGUSOV, I.S.A. On the problem of prime-twins (in Russian). Jaroslav. Gos. Ped. Inst. Ucen. Zap., 82, 1971, 85-86. 1971 TURAN, P. On some recent results in the analytical theory of numbers. Proc. Symp. Pure Mathematics (1969 Number Theory Institute), vol. 20, 359-374. Amer. Math. Soc., Providence, RI., 1971. 1972 BATEMAN, P.T. The distribution of values of Euler function. Acta Arithm., 21, 1972,329-345. 1972 DESHOUILLERS, J.M. Nombres premiers de la forme [n c]. c.R. Acad. Sci. Paris, Ser. A, 282, 1976, 131-133. Chapter 4 483

1972 HUXLEY, M.N. On the difference between consecutive primes. Invent. Math., 15, 1972, 164-170. 1972 HUXLEY, M.N. The Distribution of Prime Numbers. Oxford University Press, Oxford, 1972. 1972 ROTKIEWICZ, A. On a problem of VI/. Sierpiilski. Elem. d. Math., 27, 1972,83-85. 1972 WALL, e.R. Density bounds for Euler's function. Math. Comp., 26, 1972,779-783. 1973 APOSTOL, T.M. Another elementary proof of Euler's formula for ((2n). Amer. Math. Monthly, 80, 1973, 425- 431. 1973 BRENT, R.P. The first occurrence of certain large prime gaps. Math. Comp., 35, 1980, 1435-1436. 1973/1978 CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II. Sci. Sinica, 16,1973,157-176; and 21, 1978,421-430. 1973 HENSLEY, D. and RICHARDS, I. Primes in intervals. Acta Arithm., 25, 1973/4,375-391. 1973 MONTGOMERY, H.L. The pair correlation of zeros of the zeta function. Analytic Number Theory (Proc. Symp. Pure Math., Vol. XXIV, St. Louis, 1972), 181-193. Amer. Math. Soc., Providence, R.I., 1973. 1973 WUNDERLICH, M.e. On the Gaussian primes on the line Im(x) = 1. Math. Comp., 27, 1973,399-400. 1974 AYOUB, R.G. Euler and the zeta function. Amer. Math. Monthly, 81, 1974, 1067-1086. 1974 BRENT, R.P. The distribution of small gaps between successive primes. Math. Comp., 28, 1974,315-324. 1974 EDWARDS, H.M. Riemann's Zeta Function. Academic Press, New York, 1974. 1974 HALBERSTAM, H. and RICHERT, H.E. Sieve Methods. Academic Press, New York, 1974. 1974 LEVINSON, N. More than one third of zeros of Riemann's zeta function are on (j = -i. Adv. in Math., 13, 1984, 383-436. 1974 M~KOWSKI, A. On a problem of Rotkiewicz on pseudo-primes. Elem. d. Math., 29,1974,13. 484 Bibliography

1974 SHANKS, D. and WRENCH, J.W. Brun's constant. Math. Camp., 28,1974,293-299. 1974 THOMAS, H.E. The Waring problem for 22 bi­ quadrates. Trans. Amer. Math. Soc., 193, 1974,427-430. 1975 BRENT, R.P. Irregularities in the distribution of primes and twin primes. Math. Camp., 29, 1975,43-56. 1975 MONTGOMERY, H.L. and VAUGHAN, R.C. The exceptional set in Goldbach's problem. Acta Arithm., 27, 1975,353-370. 1975 RAM MURTY, P.M. On the Existence of "Euclidean Proofs" of Dirichlet's Theorem on Primes in Arithmetic Progressions. B.Sc. Thesis, Carleton University, Ottawa, 1975, 39 pages. 1975 ROSS, P.M. On Chen's theorem that each large even number has the form Pi + pz or Pi + PZP3. J. London Math. Soc., (2),10, 1975, 500-506. 1975 ROSSER, J.B. and SCHOENFELD, L. Sharper bounds for Chebyshev functions 8(x) and ljJ(x). Math. Camp., 29, 1975,243-269. 1975 SWIFT, J.D. Table of Carmichael numbers to 109 . Math. Camp., 29, 1975,338-339. 1976 APOSTOL, T.M. Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976. 1976 BRENT, R.P. Tables concerning irregularities in the distribution of primes and twin primes to 1011. Math. Camp., 30, 1976, 379. 1976 GERIG, S. A simple proof of the prime number theorem. J. Nb. Th., 8, 1976, 131-136. 1976 NIVEN, I. and POWELL, B. Primes in certain arithmetic progressions. Amer. Math. Monthly, 83,1976,467-469. 1976 SCHOENFELD, L. Sharper bounds for Chebyshev functions 8(x) and ljJ(x), II. Math. Camp., 30, 1976, 337-360. 1976 VAUGHAN, R.C. A note on Schnirelman's approach to Goldbach's problem. Bull. London Math. Soc., 8, 1976, 245-250. 1977 DESHOUILLERS, J.M. Sur la con stante de Schnirel­ man. Sem. Delange-Pisot-Poitou, 1Y an nee, 1975/6, fasc. 2, expo No. G16, 6 p., Paris, 1977. Chapter 4 485

1977 HUDSON, R.H. A formula for the exact number of primes below a given bound in any arithmetic progression. Bull. Austral. Math. Soc., 16,1977,67-73. 1977 HUDSON, R.H. and BRAUER, A. On the exact number of primes in the arithmetic progressions 4n ± 1 and 6n ± 1. Journal f d. reine u. angew, Math., 291, 1977, 23-29. 1977 HUXLEY, M.N. Small differences between consecutive primes, II. Mathematika, 24, 1977, 142-152. 1977 JUTILA, M. On Linnik's constant. Math. Scand., 41, 1977,45-62. 1977 JUTILA, M. Zero-density estimates for L-functions. Acta Arithm., 32, 1977, 52-62. 1977 KUMAR MURTY, V. The Least Prime in an Arithmeti­ cal Progression and an Estimate of Linnik's Constant. B.Sc. Thesis, Carleton University, Ottawa, 1977,45 pages. 1977 LANGEVIN, M. Methodes elementaires en vue du theoreme de Sylvester. Sem. Delange-Pisot-Poitou, 17 e annee, 1975/76, fasc. 2, expo No. G2, 9 pages, Paris, 1977. 1977 POWELL, B. Proof of a special case of Dirichlet's theorem. Fibonacci Quart., 15, 1977, 167-169. 1977 SMALL, C. Waring's problem. Math. M'ag., 50, 1977, 12-16. 1977 WEINTRAUB, S. Seventeen primes in arithmetic progression. Math. Comp., 31, 1977, 1030. 1977 ZAGIER, D. The first 50 million prime numbers. Math. Intelligencer, Vol. 0, 1977,7-19. Reprinted in German in Lebendige Zahlen, by W. Borho, J.e. Jantzen, H. Kraft, J. Rohlfs, D. Zagier. Birkhauser, Basel, 1981. 1978 BAYS, C. and HUDSON, R.H. Details of the first region of integers x with n 3 • 2 (x} < n 3 • 1 (x}. Math. Comp., 32, 1978, 571-576. 1978 BAYS, C. and HUDSON, R.H. On the fluctuations of Littlewood for primes of the form 4n ± 1. Math. Comp., 32, 141,281-286. 1978 ELLISON, W.J. and ELLISON, F. Theorie des Nombres. Abrege d'Histoire des Mathematiques, Vol. I, Chapter V, §VI (edited by J. Dieudonne). Hermann, Paris, 1978. 486 Bibliography

1978 HEATH-BROWN, D.R. Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Phil. Soc., 83, 1978,357-375. 1978 IWANIEC, H. On the problem of lacobsthal. Demo. Math., 11,1978,225-231. 1978 WAGSTAFF, Jr., S.S. The least prime in arithmetic progression with prime difference. Journal f d. reine u. angew. Math., 301,1978,114-115. 1979 ATKIN, A.O.L. and RICKERT, N.W. On a larger pair of twin primes. Abstract 79T-A132, Notices Amer. Math. Soc., 26, 1979, A-373. 1979 BALASUBRAMANIAN, R. On Waring's problem: g(4) ::;; 21. Hardy-Ramanujan J., 2, 1979,31 pages. 1979 BAYS, C. and HUDSON, R.H. Numerical and graph­ ical description of all axis crossing regions for the moduli 4 and 8 which occur before 10 12. Intern. J. Math. Math. Sci., 2,1979,111-119. 1979 CHEN, J.R. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet's L-functions, II. Sci. Sinica, 22, 1979, 859-889. 1979 ELLIOTT, P.D.T.A. Probabilistic Number Theory (in 2 volumes). Springer-Verlag, New York, 1979. 1979 GROSSW ALD, E. and HAGIS, Jr., P. Arithmetic progressions consisting only of primes. Math. Comp., 33, 1979, 1343-1352. 1979 HEATH-BROWN, D.R. and IWANIEC, H. On the difference between consecutive powers. Bull. Amer. Math. Soc., N.S., 1, 1979, 758-760. 1979 HLAWKA, E. Theorie der Gleichverteilung. Biblio­ graphisches Institut, Zurich, 1979. 1979 IW ANIEC, H. and JUTILA, M. Primes in short intervals. Arkiv f Mat., 17, 1979, 167-176. 1979 POMERANCE, C. The prime number graph. Math. Comp., 33, 1979, 399-408. 1979 ROTKIEWICZ, A. and WASEN, R. On a number­ theoretical series. Publ. Math. Debrecen, 26, 1979, 1-4. 1979 WAGSTAFF, Jr., S.S. Greatest of the least primes in arithmetic progressions having a given modulus. Math. Comp., 33, 1979, 1073-1080. Chapter 4 487

1979 WOOLDRIDGE, K. Values taken many times by Euler's phi-function. Proc. Amer. Math. Soc., 76, 1979, 229-234. 1980 BRENT, R.P. The first occurrence of certain large prime gaps. Math. Comp., 35, 1980, 1435-1436. 1980 CHEN, J.R. and PAN, C.D. The exceptional set of Goldbach numbers, I. Sci. Sinica, 23, 1980,416-430. 1980 ERDOS, P. and STRAUS, E.G. Remarks on the difference between consecutive primes. Elem. d. Math., 35, 1980, 115-118. 1980 IVIC, A. Exponent pairs and the zeta-function of Riemann. Studia Sci. Math. Hung., 15, 1980, 157-181. 1980 LIGHT, W.A., FORREST, J., HAMMOND, N., and ROE, S. A note on Goldbach's conjecture. BIT, 20, 1980, 525. 1980 NEWMAN, D.J. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87,1980,693-696. 1980 PINTZ, J. On Legendre's prime number formula. Amer. Math. Monthly, 87,1980, 733-735. 1980 POMERANCE, C. Popular values of Euler's function. Mathematika, 27, 1980,84-89. 1980 POMERANCE, C. A note on the least prime in an arithmetic progression. J. Nb. Th., 12, 1980.,218-223. 1980 POMERANCE, c., SELFRIDGE, J.I"., and W AG­ STAFF, Jr., S.S. The pseudo primes to 25 '109 . Math. Comp., 35, 1980, 1003-1026. 1980 POWELL, B. Problem E2844 (Difference between consecutive primes). Amer. Math. Monthly, 87, 1980, 577; 90, 1983, 286. 1980 VANDEN EYNDEN, C. Proofs that L lip diverges. Amer. Math. Monthly, 87, 1980,394-397. 1980 VAN DER POORTEN, A.J. and ROTKIEWICZ, A. On strong pseudoprimes in arithmetic progressions. J. Austral. Math. Soc., A, 29, 1980,316-321. 1980 WAGSTAFF, Jr., S.S. Greatest of the least primes in arithmetic progressions having a given modulus. Math. Comp., 33, 1979, 1073-1080. 1981 BOHMAN, J. and FROBERG, c.E. Numerical investi­ gations of Waring's problem for cubes. BIT, 21,1981,118-122. 488 Bibliography

1981 GRAHAM, S.S.W. On Linnik's constant. Acta Arithm., 39, 1981, 163-179. 1981 HEATH-BROWN, D.R. Three primes and an almost prime in arithmetic progression. J. London Math. Soc., (2), 23, 1981,396-414. 1981 LEAVITT, W.G. and MULLIN, A.A. Primes differing by a fixed integer. Math. Camp., 37, 1981,581-585. 1981 MAIER, H. Chains of large gaps between consecutive primes. Adv. in Math., 39, 1981,257-269. 1981 PINTZ, J. On primes in short intervals, I. Studia Sci. Math. Hung., 16, 1981,395-414. 1981 POMERANCE, C. On the distribution of pseudo­ primes. Math. Camp., 37, 1981,587-593. 1981 WEINTRAUB, S. A large prime gap. Math. Camp., 36, 1981,279. 1982 DIAMOND, H.G. Elementary methods in the study of the distribution of prime numbers. Bull. Amer. Math. Soc., 7, 1982, 553-589. 1982 GROSSW ALD, E. Arithmetic progressions that consist only of primes, J. Nb. Th., 14, 1982, 9-3l. 1982 NAIR, M. On Chebyshev type inequalities for primes. Amer. Math. Monthly, 81, 1982, 126-129. 1982 POMERANCE, C. A new lower bound for the pseudo-primes counting function. Illinois J. Math., 26, 1982,4-9. 1982 PRITCHARD, P.A. 18 primes in arithmetic progres­ sion. J. Recr. Math., 15, 1982/3,288. 1982 ROMANI, F. Computations concerning Waring's prob­ lem for cubes. Calcolo, 19, 1982, 415-43l. 1982 THANIGASALAM, K. Some new estimates for G(k) in Waring's problem. Acta Arithm., 42, 1982/3, 73-78. 1982 WEINTRAUB, S. A prime gap of 682 and a prime arithmetic sequence. BIT, 22, 1982, 538. 1983 CHEN, J.R. The exceptional value of Goldbach numbers, II. Sci. Sinica, Ser. A, 26, 1983, 714-73l. 1983 CONREY, J.B. Zeros of derivatives of Riemann' xi-function on the critical line. J. Nb. Th., 16, 1983,49-74. 1983 FOUVRY, E. and IWANIEC, H. Primes in arithmetic progressions. Acta Arithm., 42, 1983, 197-218. Chapter 4 489

1983 IVIC, A. Topics in Recent Zeta Function Theory. Pub!. Math. d'Orsay, Univ. Paris-Sud. 1983. 1983 KELLER, W. Large twin primes pairs related to Mersenne numbers. Abstracts Amer. Math. Soc., 4, 1983, 490. 1983 NICOLAS, J.L. Petites valeurs de la fonction d'Euler. J. Nb. Th., 17, 1983,375-388. 1983 NICOLAS, J.L. Distribution de valeurs de la fonction d'Euler. In Algorithmique, Calcul Formel Arithmetique, expose 24, 7 pages. Univ. Saint-Etienne, 1983. Reprinted in L'Enseign. Math. 30,1984,331-338. 1983 POWELL, B. Problem 6429 (Difference between con­ secutive primes). Amer. Math. Monthly, 90, 1983,338. 1983 RIESEL, H. and VAUGHAN, R.C. On sums of primes. Arkiv f Mat., 21, 1983,45-74. 1983 ROBIN, G. Estimation de la fonction de Tschebychef () sur Ie k-ieme nombre premier et grandes valeurs de la fonction w(n), nombre de diviseurs premiers de n. Acta Arithm., 42, 1983, 367-389. 1984 BALASUBRAMANIAN, R. and MOZZOCHI, c.J. An improved upper bound for G(k) in Waring's problem for relatively small k. Acta Arithm., 63, 1984,283-285. 1984 DABOUSSI, H. Sur Ie theoreme des nombres premiers. CR. Acad. Sci. Paris, Ser. I, 298, 1984, 161--164. 1984 DAVIES, R.O. Solution of problem 6429. Amer. Math. Monthly, 91, 1984,64. 1984 GUPTA, R. and RAM MURTY, P.M. A remark on Artin's conjecture. Invent. Math., 78, 1984, 127-130. 1984 IWANIEC, H. and PINTZ, J. Primes in short intervals. Monatsh. Math., 98, 1984, 115-143. 1984 PINTZ, J. On primes in short intervals, II. Stud. Sci. Math. Hung., 19, 1984,89-96. 1984 PINTZ, J. On the remainder term of the prime number formula and the zeros of Riemann's zeta function. Proc. Journees Arithmetiques, Nordwijkerhout, 1983). Lect. Notes in Math. 1968, 186-197. Springer-Verlag, Berlin, 1984. 1984 POWELL, B. and SHAFER, R.E. Solution of problem E 2844. Amer. Math. Monthly, 91, 1984,310-311. 490 Bibliography

1984 SCHROEDER, M.R. Number Theory in Science and Communication. Springer-Verlag, New York, 1984. 1984 TUR.\.N, P. A New Method in Analysis and its Applications. Wiley-Interscience, New York, 1984. 1984 WANG, Y. Goldbach Conjecture. World Scientific Publ., Singapore, 1984. 1985 BALASUBRAMANIAN, R., CONREY, J.B., and HEATH-BROWN, D.R. Asymptotic mean square ofthe product of the Riemann zeta-function and a Dirichlet polynomial. Journal f d. reine u. angew. Math., 357, 1985, 161-181. 1985 COST A PEREIRA, N. Estimates for the Chebyshev function tf;(x) - ()(x). Math. Comp., 44, 1985,211-221. 1985 FOUVRY, E. Theoreme de Brun-Titchmarsh, applica­ tion au theoreme de Fermat. Invent. Math., 79, 1985, 383- 407. 1985 HUDSON, R.H. Averaging effect on irregularities in the distribution of primes in arithmetic progressions. Math. Comp., 44, 1985,561-571. 1985 IVIC, A. The Riemann Zeta-Function. J. Wiley and Sons, New York, 1985. 1985 LAGARIAS, J.e., MILLER, V.S., and ODLYZKO, A.M. Computing n(x): The Meissel-Lehmer method. Math. Comp., 44,1985,537-560. 1985 MAIER, H. Primes in short intervals. Michigan Math. J.,32, 1985,221-225. 1985 ODLYZKO, A.M. and TE RIELE, H.J.J. Disproof of the Mertens conjecture. Journal f d. reine u. angew. Math., 357, 1985, 138-160. 1985 POWELL, B. Problem 1207 (A generalized weakened Goldbach theorem). Math. Mag., 58, 1985, 46; 59, 1986, 48-49. 1985 PRITCHARD, P.A. Long arithmetic progressions of pnmes; some old, some new. Math. Comp., 45, 1985, 263-267. 1985 TE RIELE, H.J.J. Some historical and other notes about Mertens conjecture and its recent disproof. Nieuw Arch v. Wisk. (4) 3, 1985,237-243. Chapter 4 491

1985 THANIGASALAM, K. Improvement on Davenport's iterative method and new results in additive number theory, I and II (Proof that G(5) ~ 22). Acta Arithm., 46, 1985,1-31 and 91-112. 1986 BALASUBRAMANIAN, R., DESHOUILLERS, J.M., and DRESS, F. Probleme de Waring pour les bicarrt!s, 2: resultats auxiliaires pour Ie theoreme asymptotique. c.R. Acad. Sci. Paris, Ser. I, 303, 1986, 161-163. 1986 BOMBIERI, E. and IWANIEC, E. On the order of ((t + it). Ann. Sci. N. Sup. Pisa (4),13, 1986,449-472. 1986 BOMBIERI, E., FRIEDLANDER, J.B., and IWANIEC, H. Primes in arithmetic progression to large moduli, I. Acta Math., 156, 1986,203-251. 1986 COSTA PEREIRA, N. Sharp elementary estimates for the sequence of primes. Port. Math., 43, 1986, 399-406. 1986 FINN, M.V. and FROHLIGER, J.A. Solution of problem 1207. Math. Mag., 59, 1986,48-49. 1986 MOZZOCHI, c.J. On the difference between consecu­ tive primes. J. Nb. Th., 24, 1986, 181-187. 1986 PINTZ, J. A note on the exceptional set in Goldbach's problem. Math. Institute Hungarian Acad. Sci., Preprint No. 14/1986. 1986 TE RIELE, H.J.J. On the sign of the difference n(x) - li(x). Report NM-R8609, Centre for Math. and Compo Science, Amsterdam, 1986; Math. Comp., 48, 1987, 323-328. 1986 VAN DE LUNE, J., TE RIELE, H.J.J. and WINTER, D.T. On the zeros of the Riemann zeta function in the critical strip, IV. Math. Comp., 46, 1986,667-·681. 1986 VAUGHAN, R.C. On Waring's problem for small exponents. Proc. London Math. Soc., 52, 1986,445-463. 1986 VAUGHAN, R.C. On Waring's problem for sixth powers. J. London Math. Soc., (2), 33, 1986,227-236. 1986 WAGON, S. Where are the zeros of zeta of s? Math. Intelligencer 8, No.4, 1986,57-62. 1987 BOMBIERI, E., FRIEDLANDER, J.B., and IWANIEC, H. Primes in arithmetic progressions to large moduli, II. Math. Ann., 277, 1987,361-393. 492 Bibliography

1987 BOMBIERI, E. and IWANIEC, H. On the order of ((t + it). Ann. Scuola Norm. Sup. Pisa. 1987 GIORDANO, G. The generalization and proof of Bertrand's postulate. Intern. J. Math. Math. Sci., 10, 1987, 821-824. 1987 ODLYZKO, A.M. On the distribution of spacings between zeros of the zeta function. Math. Comp., 48, 1987, 273-308. 1987 PINTZ, J. An effective disproof of the Mertens conjecture. Asterisque, 147/148, 1987, 325-333. 1987 THANIGASALAM, K. Improvement on Davenport's iterative method and new results in additive number theory, III. Acta Arithm., 48, 1987,97-116. 1988 ERDOS, P., KISS, P. and SARKOZY, A. A lower bound for the counting function of Lucas pseudoprimes. Math. Comp., 51, 1988,315-323. 1988 HILDEBRAND, A. and MAIER, H. Gaps between prime numbers. Proc. Amer. Math. Soc., 104,1988,1-10. 1988 MAIER, H. Small differences between primes. Michigan Math. J. 35, 1988,323-344. 1988 PATTERSON, S.J. Introduction to the Theory oj the Riemann Zeta-Junction. Cambridge University Press, Cambridge, 1988. 1989 CHEN, J.R. and WANG, Y. On the odd Goldbach problem. Acta Math. Sci. Sinica, 32, 1989, 702-718. 1989 CONREY, J.B. At least two-fifths of the zeros of the Riemann zeta function are on the critical line. Bull. Amer. Math. Soc., 20, 1989, 79-81. 1989 GIORDANO, G. Further results on primes in small intervals, Intern. J. Math. Math. Sci., 12,1989,441-446. 1989 GIORDANO, G. The existence of primes in small intervals. Indian J. Math., 31, 1989, 105-110. 1989 GORDON. D.M. On the number of elliptic pseudo­ primes. Math. Comp., 52,1989,231-245. 1989 GRANVILLE, A., V AN DE LUNE, J., and TE RIELE, H.J.J. Checking the Goldbach's conjecture on a vector computer. In Number Theory and Applications (edited by R.A. Mollin), 423-433. Kluwer, Dordrecht., 1989. 1989 MAIER, H. and POMERANCE, C. Unusually large Chapter 4 493

gaps between consecutive primes. In Theorie des N ombres­ Number Theory (Proc. Intern. Number Th. eonf., Quebec, 1987, editors I.M. de Koninck, C. Levesque), 625-632. W. de Gruyter, Berlin, 1989. 1989 MIYAMOTO, J. and RAM MURTY, P.M. Elliptic pseudoprimes. Math. Comp., 53, 1989,415-430. 1989 YOUNG, J., and POTLER, A. First occurrence of prime gaps. Math. Comp., 52,1989,221-224. 1990 DELMER, F. and DESHOUILLERS, J.M. On the computation of g(k) in Waring's problem. Math. Comp., 54, 1990,885-893. 1990 FELGNER, U. Estimates for the sequence of primes. Elem. d. Math., 46, 1990, 17-25. 1990 GRANVILLE, A. and POMERANCE, C. On the least prime in certain arithmetic progressions. J. London Math. Soc., (2), 41, 1990, 193-200. 1990 HEATH-BROWN, D.R. The difference between con­ secutive primes, IV. In A Tribute to Paul Erdos (edited by A. Baker, B. Bollobas, and A. Hajnal) 277-287. Cambridge University Press, Cambridge, 1990. 1990 JAESCHKE, G. The Carmichael numbers to 1012• Math. Comp., 55, 1990,383-389. 1990 KUBINA, J.M. and WUNDERLICH, M.C. Extending Waring's conjecture to 471,600,000. Math. Comp., 55, 1990, 815-820. 1990 MAIER, H. and POMERANCE, C. Unusually large gaps between consecutive primes. Trans. Amer. Math. Soc., 322, 1990,201-237. 1990 PARADY, B.K., SMITH, J.F., and ZARANTONELLO, S. Largest known twin primes. Math. Comp., 55, 1990, 381-382. 1991 GORDON, D.M. and POMERANCE, C. The distribu­ tion of Lucas and elliptic pseudoprimes. Math. Comp., 57, 1991, 825-838. 1991 WEINTRAUB, S. A prime gap of 784. J. Recr. Math., 23, 1991,6-7. 1992 DESHOUILLERS, J.M. and DRESS, F. Sum of 19 biquadrates: on the representation of large integers. Anrc. Scuola Norm. Sup. Pisa, CI. Sci., (4), 92, 1992, 113-153. 494 Bibliography

1992 GRANVILLE, A. Primality testing and Carmichael numbers. Notices Amer. Math. Soc., 39,1992,696-700. 1992 HEATH-BROWN, D.R. Zero-free regions for Diri­ chlet L-functions and the least prime in an arithmetic progression. Proc. London Math. Soc., (3), 64, 1992, 265- 338. 1993 DAMGARD, I., LANDROCK, P. and POMERANCE, C. Average case error estimates for the strong probable prime test. Math. Comp., 61,1993,177-194. 1993 DESHOUILLERS, J.M. and DRESS, F. Numerical results for sums of five and seven biquadrates and con­ sequences for sums of 19 biquadrates. Math. Comp., 61, 1993, 195-207. 1993 DESHOUILLERS, J.M., GRANVILLE, A., NARKIE­ WICZ, W., and POMERANCE, C. An upper bound in Goldbach's problem. Math. Comp., 61, 1963,209-213. 1993 LOU, S. and YAO, Q. The number of primes in a short interval. Hardy-Ramanujan J., 16,1993,21-43. 1993 ODLYZKO, A. Iterated absolute values of differences of consecutive primes. Math. Comp., 61, 1993,373-380. 1993 PINCH, R.G.E. The Carmichael numbers up to 1015. Math. Comp., 61, 1993, 703-722. 1993 POMERANCE, C. Carmichael numbers. Nieuw Arch. v. Wzsk., (4), 11, 1993, 199-209. 1993 SINISALO, M.K. Checking the Goldbach's conjecture up to 4 X 1011. Math. Comp., 61, 1993,931-934. 1993 WEINTRAUB, S. Consecutive primes in arithmetic progression. J. Recr. Math., 25,1993,164-166. 1993 WEINTRAUB, S. A prime gap of 864. J. Recr. Math., 25, 1993,42-43. 1994 DELEGLISE, M. and RIVAT, J. Computing n(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method. Pre­ print (1994). 1994 LOU, S. and YAO, Q. Estimates of sums of Dirichlet series, Hardy-Ramanujan J., 17, 1994, 1-31. 1994 ODLYZKO, A.M. Analytic computations in number theory. In Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics (ed. N. Chapter 5 495

Gautschi), 451-463. Proc. Symp. Appl. Math., # 48, Amer. Math. Soc., Providence, R.I., 1994. 1994 RAMARE, O. On Schnirelman's constant. Preprint. 1995 BAKER, R.C. and HARMAN, G. The difference between consecutive primes, Proc. London At ath. Soc. (3), 71, 1995 (to appear). 1995 GRAHAM, S.W. Carmichael numbers with three prime factors (to appear). 1995 INDLEKOFER, K.-H. and JARAI, A. Largest twin primes, Math. Compo (to appear in 1995). 1995 KUTRIB, M. and BICHSTEIN, J. On the number of twin primes and some other constellations of primes less than a given limit. Preprint (1995). 1995 NICELY, T.R. Enumeration to 10 14 of the twin primes and Brun's constant. Preprint (1995). 1995 PRITCHARD, P., MORAN, A., and THYSSEN, A. Twenty-two primes in arithmetic progression, Math. Comp., 64, 1995, 1337-1339.

CHAPTER 5 1878 LUCAS, E. Theorie des fonctions numeriques simple­ ment periodiques. Amer. J. Math., 1, 1878, 185-240 and 289-321. 1902 BACHMANN, P. Niedere Zahlentheorie, Vol. II. Teub­ ner, Leipzig, 1902. Reprinted by Chelsea, Bronx, N.Y., 1968. 1917 POLLACZEK, F. Uber den grossen Fermat'schen Satz. Sitzungsber. Akad. d. Wiss. Wien, Abt. IIa, 126, 1917,45-59. 1937 HALL, Jr., M. Divisors of second-order sequences. Bull. Amer. Math. Soc., 43, 1937, 78-80. 1940 KRASNER, M. A propos du critere de Sophie Germain-Furtwangler pour Ie premier cas clu theoreme de Fermat. Mathematica Cluj, 16, 1940, 109-114. 1948 GUNDERSON, N.G. Derivation of Criteria for the First Case of Fermat's Last Theorem and the Combination of these Criteria to Produce aNew Lower Bound for the Exponent. Ph.D. Thesis, Cornell University, 1948, 111 pages. 496 Bibliography

1951 DENES, P. An extension of Legendre's criterion in connection with the first case of Fermat's last theorem. Pub!. Math. Debrecen, 2, 1951, 115-120. 1952 ERDOS, P. and MIRSKY, L. The distribution of values of the divisor function d(n). Proc. London Math. Soc., (3), 2, 1952,257-271. 1953 GOLDBERG, K. A table of Wilson quotients and the third Wilson prime. J. London Math. Soc., 28, 1953, 252-256. 1954 WARD, M. Prime divisors of second order recurring sequences. Duke Math. J., 21, 1954,607-614. 1956 OBLATH, R. Une propriete des puissances parfaites. Mathesis,65, 1956,356-364. 1956 RIESEL, H. Nigra stora primtal. Elementa, 39, 1956, 258-260. 1958 SIERPINSKI, W. Sur une decomposition des nombres premiers en deux classes. Collect. Math., 10,1958,81-83. 1959 SIERPINSKI, W. Sur les nombres premiers ayant des chiffres initiaux et finals donnes. Acta Arithm., 5, 1959, 265-266. 1960 BRAUER, A. A note on a number theoretical paper of Sierpinski. Proc. Amer. Math. Soc., 11, 1960,406-409. 1960 SIERPINSKI, W. Sur un probleme concernant les nombres k· 2n + 1. Elem. d. Math., 15, 1960, 73-74. 1960 WALL, D.D. Fibonacci series modulo m. Amer. Math. Monthly, 67, 1960,525-532. 1961 AIGNER, A. Folgen der Art arn + b, welche nur teilbare Zahlen liefern. Math. Nachr., 23, 1961,259-264. 1961 WARD, M. The prime divisors of Fibonacci numbers. Pacific J. Math., 11, 1961,379-386. 1963 BRILLHART, J. Some miscellaneous factorizations. Math. Comp., 17, 1963,447-450. 1963 PEARSON, E.H. On the congruences (p - 1)! == -1 and 2P-l == 1 (mod p2). Math. Comp., 17,1963,194-195. 1963 SELFRIDGE, J.L. Solution to problem 4995 (proposed by O. Ore). Amer. Math. Monthly, 70,1963,101. 1963 VOROB'EV, N.N. Fibonacci Numbers. Heath & Co., Boston, 1963. Chapter 5 497

1964 GRAHAM, R.L. A Fibonacci-like sequence of compos­ ite numbers. Math. Mag., 37,1964,322-324. 1964 SIEGEL, C.L. Zu zwei Bemerkungen Kummers. Nachr. Akad. d. Wiss. Gottingen, Math. Phys. KI., II, 1964, 51-62. Reprinted in Gesammelte Abhandlunqen (edited by K. Chandrasekharan and M. MaaB), Vol. III, 436-442. Springer-Verlag, Berlin, 1966. 1965 KLOSS, K.E. Some number theoretic calculations. J. Res. Nat. Bureau of Stand., B, 69, 1965, 335-336. 1965 ROTKIEWICZ, A. Sur les nombres de Mersenne depourvus de diviseurs carres et sur les nombres naturels n tels que n2 12" - 2. Mathem. Vecnik, (2), 17, 1965, 78- 80. 1966 HASSE, H. Uber die Dichte der Primzahlen p, fur die eine vorgegebene ganzrationale Zahl a =1= 0 von gerader bzw. ungerader Ordnung mod p ist. Math. Ann., 168, 1966, 19-23. 1966 KRUYSWIJK, D. On the congruence Up-l == 1 (mod p2) (in Dutch). Math. Centrum Amsterdam, 1966, 7 pages. 1967 WARREN, L.J. and BRAY, H. On the square-freeness of Fermat and Mersenne numbers. Pacific J. Math., 22, 1967,563-564. 1968 PUCCIONI, S. Un teorema per una resoluzione parziale del famoso teorema di Fermat. Archimede, 20, 1968, 219-220. 1970 GOLOMB, S.W. Powerful numbers. Amer. Math. Monthly, 77, 1970, 848-852. 1971 BRILLHART, J., TONASCIA, J., and WICINBERGER, P.J. On the Fermat quotient. Computers in Number Theory (edited by A.L. Atkin and B.l. Birch), 213-222. Academic Press, New York, 1971. 1971 UCHIDA, K. Class numbers of imaginary abelian number fields, III. Tohoku Math. J., 23, 1971, 573- 580. 1972 IWASAWA, K. Lectures on p-adic L-functions. Annals of Math. Studies, Princeton Univ. Press, Princeton, 1972. 1972 MASLEY, J.M. On the Class Number of Cyclotomic Fields. Ph.D. Thesis, Princeton Univ., 1972, 51 pages. 498 Bibliography

1973 VAUGHAN, R.C. A remark on the divisor function d(n). Glasgow Math. J., 14, 1973, 54-55. 1974 ANGELL, 1.0. and GODWIN, H.J. Some factoriza­ tions of lOn ± 1. Math. Comp., 28, 1974,307-308. 1974 BORUCKI, L.J. and DIAZ, J.B. A note on primes, with arbitrary initial or terminal decimal ciphers in Dirichlet arithmetic progressions. Amer. Math. Monthly, 81, 1974, 1001-1002. 1974 HALBERSTAM, H. and RICHERT, H.E. Sieve Methods. Academic Press, New York, 1974. 1975 ERDOS, P. Problems and results on consecutive integers. Eureka, 38, 1975/6, 3-8. 1975 JOHNSON, W. Irregular primes and cyclotomic invariants. Math. Comp., 29, 1975, 113-120. 1975 METSANKYLA, T. On the cyclotomic invariants of Iwasawa. Math. Scand., 37, 1975,61-75. 1976 ERDOS, P. Problems and results on consecutive inte­ gers. Pub I. Math. Debrecen, 23, 1976, 271-282. 1976 HOOLEY, C. Applications of Sieve Theory of Numbers, Cambridge Tracts in Math., 70, Cambridge University Press, 1976. 1976 MASLEY, J.M. and MONTGOMERY, H.L. Unique factorization in cyclotomic fields. Journal f d. reine u. angew. Math., 286/7, 1976,248-256. 1976 MENDELSOHN, N.S. The equation ~(x) = k. Math. Mag., 49, 1976,37-39. 1976 STEPHENS, P.J. Prime divisors of second order linear recurrences, I, II. J. Nb. Th., 8, 1976,313-345. 1977 JOHNSON, W. On the non-vanishing of Fermat's quotient (mod p). Journal f d. reine u. angew. Math., 292, 1977, 196-200. 1977 POMERANCE, C. On composite n for which ~(n)ln - 1, II. Pacific J. Math., 69,1977,177-186. 1977 POWELL, B. Problem E 2631 (Prime satisfying Mirimanoffs condition). Amer. Math. Monthly, 84, 1977,57. 1978 DE LEON, M.J. Solution of problem E 2631. Amer. Math. Monthly, 85, 1978,279-280. 1978 METSANKYLA, T. Iwasawa invariants and Kummer congruences. J. Nb. Th., 10, 1978,510-522. Chapter 5 499

1978 WAGSTAFF, Jr., S.S. The irregular primes to 125000. Math. Camp., 32, 1978, 583-591. 1978 WILLIAMS, H.C. Some primes with interesting digit patterns. Math. Comp., 32, 1978, 1306-1310. 1979 BAYER, P. Sobre el indice de irregularidad de los numeros primos. Collect. Math., 30, 1979, 11-20. 1979 ERDOS, P. and ODLYZKO, A.M. On the density of odd integers of the form (p - l)rn and related questions. J. Nb. Th., 11, 1979,257-263. 1979 FERRERO, B. and WASHINGTON, L.C. The Iwasawa

invariant J1p vanishes for abelian number fields. Ann. Math., 109, 1979,377-395. 1979 RIBENBOIM, P. 13 Lectures on Fermat's Last Theo­ rem. Springer-Verlag, New York, 1979. 1979 WILLIAMS, H.C. and SEAH, E. Some primes of the form (an - l)j(a - 1). Math. Camp., 23, 1979, 1337- 1342. 1980 NEWMAN, M., SHANKS, D. and WILLIAMS, H.C. Simple groups of square order and an interesting sequence of primes. Acta Arithm., 38, 1980, 129-140. 1980 POWELL, B. Primitive densities of oertain sets of primes. J. Nb. Th., 12, 1980,210-217. 1980 SKULA, L. Index of irregularity of a prime. Journal f d. reine u. angew. Math., 315, 1980,92-106. 1980 WASHINGTON, L.C. Introduction to Cyclotomic Fields. Springer-Verlag, New York, 1980. 1981 LEHMER, D.H. On Fermat's quotient, base two. Math. Comp., 36, 1981,289-290. 1981 SHANKS, D. and WILLIAMS, H.C. Gunderson's function in Fermat's last theorem. Math. Comp., 36, 1981, 291-295. 1981 SPIRO, C.A. The Frequency with which an Integral­ Valued. Prime-Independent. Multiplicative or Additive Function of n Divides a Polynomial Function of n. Ph.D. Thesis, University of Illinois, Urbana-Champaign, 1981, 179 pages. 1982 POWELL, B. Problem E 2948 (p e ll x p -l - yP-l, 2% pe, p prime occurs frequently). Amer. Math. Monthly, 89, 1982, 334. 500 Bibliography

1982 POWELL, B. Problem E 2956 (The existence of small prime solutions of x p- 1 =1= 1 (mod p2)). Amer. Math. Monthly, 89, 1982, 498. 1982 WILLIAMS, H.C. The influence of computers in the development of number theory. Comp. & M aths. with Appl., 8, 1982, 75-93. 1982 WILLIAMS, H.C. A note on the Fibonacci quotient Fp-t/p. Can. Math. Bull., 25, 1982,366-370. 1982 YATES, S. Repunits and Repetends. Star Publ. Co., Boynton Beach, Florida, 1982. 1983 JAESCHKE, G. On the smallest k such that k· 2N + 1 are composite. Corrigendum. Math. Comp., 40, 1983, 381-384; 45, 1985,637. 1983 KELLER, W. Factors of Fermat numbers and large primes of the form k· 2n + 1. Math. Comp., 41, 1983, 661-673. 1983 RIBENBOIM, P. 1093. Math. Intelligencer, 5, No.2, 1983,28-34. 1984 DA VIS, J.A. and HOLDRIDGE, D.B. Most wanted factorization using the quadratic sieve. Sandia Nat. Lab. Report SAND 84-1658, 1984. 1984 HEATH-BROWN, D.R. The divisor function at con­ secutive integers. Mathematika, 31, 1984, 141-149. 1984 MATTICS, L.E. Solution of problem E 2948. Amer. Math. Monthly, 91, 1984,650-651. 1985 AD LEMAN, L.M. and HEATH-BROWN, D.R. The first case of Fermat's last theorem. Invent. Math., 79, 1985, 409-416. 1985 DUBNER, H. Generalized Fermat primes. J. Recr. Math., 18, 1985,279-280. 1985 FOUVRY, E. Theoreme de Brun-Titchmarsh; appli­ cation au theoreme de Fermat. Invent. Math., 79, 1985, 383-407. 1985 GRANVILLE, A. Refining the conditions on the Fermat quotient. Math. Proc. Cambridge Phil. Soc., 98, 1985,5-8. 1985 KELLER, W. The 17th prime of the form 5· 2n + 1. Abstracts Amer. Math. Soc., 6, No.1, 1985, 121. Chapter 5 501

1985 LAGARIAS, J.e. The set of primes dividing the Lucas number has density j. Pacific J. Math., 118, 1985, 19- 23. 1985 RIBENBOIM, P. An extension of Sophie Germain's method to a wide class of diophantine equations. Journal! d. reine u. angew. Math., 356,1985,49-66. 1986 GRANVILLE, A. Powerful numbers and Fermat's last theorem. c.R. Math. Rep. Acad. Sci. Canada, 8, 1986, 215-218. 1986 McCURLEY, K.S. Polynomials with no small prime value. Proc. Amer. Math. Soc., 97,1986,393-395. 1986 MOLLIN, R.A. and WALSH, P.G. On powerful numbers. Intern. J. Math. Math. Sci., 9, 1986,801-806. 1986 MORIMOTO, M. On prime numbers of Fermat type (in Japanese), Sugaku, 38, 1986,350-354. 1986 SKULA, L. A note on the index of irregularity. J. Nb. Th., 22, 1986, 125-138. 1986 SPIRO, e.A. An iteration problem involving the divisor function. Acta Arithm., 46, 1986, 17-27. 1986 TZANAKIS, N. Solution to problem E 2956. Amer. Math. Monthly, 93, 1986,569. 1986 WILLIAMS, H.e. and DUBNER, H. The primality of RI03l. Math. Comp., 47, 1986, 703-712. 1987 GRANVILLE, A. Diophantine Equations with Vari­ able Exponents with Special Reference to Fermat's Last Theorem. Ph.D. Thesis, Queen's University, Kingston, 1987,207 pages. 1987 ROTKIEWICZ, A. Note on the diophantine equation 1 + x + x 2 + ... + x" = ym. Elem. d. Math., 42, 1987, 76. 1987 TANNER, J.W. and WAGSTAFF, Jr., S.S. New congruences for the Bernoulli numbers. Math. Comp., 48, 1987,341-350. 1988 BRILLHART, J., MONTGOMERY, P.L. and SILVER­ MAN, R.D. Tables of Fibonacci and Lucas factoriza­ tions. Math. Comp., 50, 1988,251-260 and SI-SI5. 1988 BUELL, D.A. and YOUNG J. Some large primes and the SierpiiIski problem. SRC Techn. Rep. 88004, Super-Computing Res. Center. Lanham, MD. 502 Bibliography

1988 GONTER, R.H. and KUNDERT, E.G. Wilson's theo­ rem (mod p) (n - 1)! == -1 (mod p2), has been computed up to 10,000,000. Fourth SIAM Conference on Discrete Mathematics, San Franciso, June 19, 1988. 1988 GRANVILLE, A. and MONAGAN, M.B. The first case of Fermat's last theorem is true for all prime exponents up to 714,591,416,091,389. Trans. Amer. Math. Soc., 306, 1988, 329-359. 1988 KELLER, W. New prime solutions p of a"-l == 1 (mod p2). Abstracts Amer. Math. Soc., 9(6), 1988,503. 1988 KELLER, W. The least prime of the form k x 2" + 1 for certain values of k. Abstracts Amer. Math. Soc., 9, 1988, 417-418. 1988 RIBENBOIM, P. Impuissants devant les puissances. Expo. Math., 6, 1988,3-28. 1989 DUBNER, H. Searching for Wilson primes, J. Recr. Math., 21, 1989, 19-20. 1989 LOH, G. Long chains of nearly doubled primes. Math. Camp., 53, 1989, 751-759. 1989 TANNER, J.W. and WAGSTAFF, Jr., S.S. New bound for the first case of Fermat's last theorem. Math. Camp., 53, 1989, 743-750. 1990 COPPERSMITH, D. Fermat's last theorem (case I) and the Wieterich criterion. Math. Camp., 54, 1990, 895- 902. 1990 FUNG, G.W. and WILLIAMS, H.C. Quadratic poly­ nomials which have a high density of prime values. Math. Camp., 55, 1990, 345-353. 1991 AALTONEN, M. and INKERI, K. Catalan's equation x P - yq = 1 and related congruences. Math. Camp., 56, 1991, 359-370. Reprinted in Collected Papers of Kustaa Inkeri, (ed. by T. MetsankyUi and P. Ribenboim) Queen's Papers in Pure and Appl. Math., 91, 1992,545-558. 1991 BUHLER, J.P., CRANDALL, R.E., and SOMPOLSKI, R.W. Irregular primes to one million, Math. Camp., 59, 1992, 717-722. 1991 FEE, G. and GRANVILLE, A. The prime factors of Wendt's binomial circulant determinant (to appear). Chapter 5 503

1991 KELLER, W. Woher kommen die gr()ssten derzed bekannten Primzahlen? Mitt. Math. Ges. Hamburg, 12, 1991,211-229. 1991 SUZUKI, J. Some computations of the generalized Wieferich criteria. Preprint. 1992 KELLER, W. Factors of Fermat numbers and large primes of the form k x 2" + 1, II. Preprint (1992). 1992 SKULA, L. The Kummer's system of congruences and index of irregularity. Oster.-Ung.-Slow. Koll iiber Zahlen­ theorie Graz-Mariatrost, 1992, Grazer Math. Ber., 318, 1992, 169-172. 1993 BUHLER, J., CRANDALL, R.E. ERNVALL, R., and METSANKYLA, T. Irregular primes and cyclotomic invariants to four million. Math. Camp., 61, 1993, 151- 153. 1993 DUBNER, H. Generalized repunit primes. Math. Camp., 61, 1993,927-930. 1993 MONTGOMERY, P.L. New solutions of a P- 1 == 1 (mod p2). Math. Camp., 61,1993,361-363. 1994 DUBNER, H. Large Sophie Germain primes (to appear in Math. Camp.). 1994 DUBNER, H. Palindromic Sophie Germain primes, J. Recr. Math., 26, 1994,38-41. 1995 CRANDALL, R.E., DILCHER, K., and POMERANCE, C. A search for Wieferich and Wilson primes (to appear in Math. Camp.). 1995 DUBNER, H. and KELLER, W. Factors of generalized Fermat numbers, Math. Camp., 64,1995,397-405. 1995 DUBNER, H. and KELLER, W. New Fibonacci and Lucas primes (preliminary report). Abstracts Amer. Math. Soc., 16, 1995,267. 1995 KELLER, W. New Cullen primes (to appear in Math. Camp., 1995). 1995 KELLER, W. and NIEBUHR, W. Supplement to "New Cullen primes" (to appear in Math. Camp., 1995). 1995 KONHAUSER, J., VELLEMAN, D. and WAGON, S. lVhich Way Did the Bicycle Go? Math. Assoc. America, Do1ciani Series, Washington, 1995 (to appear). 504 Bibliography

1995 TAYLOR, R., and WILES, A. Ring-theoretic properties of certain Hecke algebras. Annals of Math. (2), 141, 1995, 553-572. 1995 WILES, A. Modular elliptic curves and Fermat's Last Theorem. Annals of Math. (2), 141, 1995,443-551.

CHAPTER 6 1857 BOUNIAKOWSKY, V. Nouveaux theoremes relatifs a la distribution des nombres premiers et a la decomposition des entiers en facteurs. M em. Acad. Sci. St. Petersbourg, (6), Sci. Math. Phys., 6, 1857,305-329. 1904 DICKSON, L.E. A new extension of Dirichlet's theorem on prime numbers. Messenger of Math., 33, 1904, 155-161. 1922 NAGELL, T. Zur Arithmetik der Polynome. Abhandl. Math. Sem. Univ. Hamburg, 1, 1922, 179-194. 1923 HARDY, G.H. and LITTLEWOOD, J.E. Some prob­ lems in "Partitio Numerorum," III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1-70. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 561-630. Clarendon Press, Oxford, 1966. 1931 HEILBRONN, H. Uber die Verteilung der Primzahlen in Polynomen. Math. Ann., 104, 1931, 794-799. 1937 BILHARZ, H. Primdivisoren mit vorgegebener Primi­ tivwurzel. Math. Ann., 114, 1937,476-492. 1937 RICCI, G. Su la congettura di Goldbach e la constante de Schnirelman. Annali Scuola Norm. Sup. Pisa, 6(2), 1937, 71-116. 1948 LIENARD, R. Tables fondamentales a 50 decimales des sommes Sn' Un' Ln' Centre de Docum. Univ., Paris, 1948. 1948 WElL, A. Sur les Courbes Algebrique et les Varietes qui s'en Deduisent. Hermann, Paris, 1948. 1950 HASSE, H. Vorlesungen iiber Zahlentheorie. Springer­ Verlag, Berlin, 1950. 1954 KUHN, P. Uber die Primteiler eines Polynoms. Proc. Intern. Congress Math., Amsterdam, 2, 1954,35-37. 1958 SCHINZEL, A. and SIERPINSKI, W. Sur certaines hypotheses concernant les nombres premiers. Remarque. Acta Arithm., 4, 1958, 185-208 and 5, 1959,259. Chapter 6 505

1960 SHANKS, D. On the conjecture of Hardy and Littlewood concerning the number of primes of the form n2 + a. Math. Comp., 14, 1960,321-332. 1961 SCHINZEL, A. Remarks on the paper "Sur certaines hypotheses concernant les nombres premiers." Acta Arithm., 7, 1961, 1-8. 1961 WRENCH, J.W. Evaluation of Artin's constant and the twin prime constant. Math. Comp., 15, 1961,396-398. 1962 BATEMAN, P.T. and HORN, R.A. A heuristic asymp­ totic formula concerning the distribution of prime numbers. Math. Comp., 16, 1962,363-367. 1964 ANKENY, N.C. and ONISHI, H. The general sieve. Acta Arithm., 10, 1964,31-62. 1964 GILLIES, D.B. Three new Mersenne primes and a statistical theory. Math. Comp., 18, 1964,93--98. 1964 SHANKS, D. An analytic criterion for the existence of infinitely many primes of the form !(n2 + 1). Illinois J. Math., 8, 1964,377-379. 1964 SIEGEL, c.L. Zu zwei Bemerkungen Kummers. N achr. Akad. d. Wiss. Gottingen, Math. Phys. ](1., II, 1964, 51-62. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan and H. MaaB), Vol III, 436-442. Springer-Verlag, Berlin, 1966. 1964 SIERPINSKI, W. Les binomes x 2 + n et les nombres premiers. Bull. Soc. Roy. Sci. Liege, 36, 1964, 259- 260. 1965 BATEMAN, P.T. and HORN, R.A. Primes represented by irreducible polynomials in one variable. Theory of Numbers (Proc. Symp. Pure Math., Vol. VIII), 119-132. Amer. Math. Soc., Providence, RI, 1965. 1966 GROSSWALD, E. Topicsfrom the Theory of Numbers. Macmillan, New York, 1966; second edition Birkhauser, Boston, 1984. 1967 HOOLEY, C. On Artin's conjecture. Journal f d. reine u. angew. Math., 225, 1967,209-220. 1967 SHANKS, D. and KRAVITZ, S. On the distribution of Mersenne divisors. Math. Comp., 21, 1967,97-101. 1969 RIEGER, G.J. On polynomials and almost-primes. Bull. Amer. Math. Soc., 75, 1969, 100-103. 506 Bibliography

1971 SHANKS, D. A low density of primes. J. Recr. Math., 4, 1971/2,272-275. 1972 RIBENBOIM, P. Algebraic Numbers. Wiley-Interscience, New York, 1972. 1973 RADEMACHER, H. Topics in Analytic Number The­ ory. Springer-Verlag, New York, 1975. 1973 WUNDERLICH, M.e. On the Gaussian primes on the line Im(x) = 1. Math. Comp., 27, 1973,399-400. 1975 SHANKS, D. Calculation and applications of Epstein zeta functions. Math. Comp., 29, 1975,271-287. 1978 IW ANIEC, H. Almost-primes represented by quadratic polynomials. I nvent. Math., 47, 1978, 171-188. 1978 SLOWINSKI, D. Searching for the 27th Mersenne prime. J. Recr. Math., 11, 1978/9,258-261. 1980 LENSTRA, Jr., H.W. Primality testing. Studieweek Getaltheorie en Computers. Stichting Math. Centrum, Amsterdam, 1980. 1981 POWELL, B. Problem 6384 (Numbers of the form mP - n). Amer. Math. Monthly, 89, 1982,278. 1982 ISRAEL, R.B. Solution of problem 6384. Amer. Math. Monthly, 90, 1983,650. 1982 SCHINZEL, A. Selected Topics on Polynomials. Univer­ sity of Michigan Press, Ann Arbor, 1982. 1983 AD LEMAN, L.M. and ODLYZKO, A.M. Irreducibil­ ity testing and factorization of polynomials. Math. Comp., 41, 1983, 699-709. 1983 SCHROEDER, M.R. Where is the next Mersenne prime hiding? Math. Intelligencer, 5, No.3, 1983,31-33. 1983 WAGSTAFF, Jr., S.S. Divisors of Mersenne numbers. Math. Comp., 40, 1983,385-397. 1984 GUPTA, R. and RAM MURTY, P.M. A remark on Artin's conjecture. Invent. Math., 78, 1984, 127-130. 1984 McCURLEY, K.S. Prime values of polynomials and irreducibility testing. Bull. Amer. Math. Soc., 11, 1984, 155-158. 1986 HEATH-BROWN, D.R. Artin's conjecture for primi­ tive roots. Quart. J. Math., Oxford, (2) 37, 1986,27-38. 1986 McCURLEY, K.S. The smallest prime value of xn + a. Can. J. Math., 38,1986,925-936. Conclusion 507

1986 McCURLEY, K.S. Polynomials with no small pnme values. Proc. Amer. Math. Soc., 97, 1986,393-395. 1987 RAM MURTY, P.M. and SRINIV ASAN, S. Some remaks on Artin's conjecture. Can. Math. Bull., 30, 1987, 80-85. 1991 MURATA, L. A problem analogous to Artin's conjec­ ture for primitive roots and its applications. Archiv. d. Math., 57, 1991,555-565. 1991 MURATA, L. The distribution of primes satisfying the condition a P == 1 (mod p2). Acta Arithm., 58, 1991, 103-111. 1993 PAPPALARDI, F. On Artin's conjecture for primitive roots. Ph.D. Thesis, McGill University, 1993.

CONCLUSION 1981 BATEMAN, P.T. Major figures in the history of the prime number theorem. Abstracts Amer. Math. Soc. (87th annual meeting, San Francisco), 1981, p. 2. 1981 BASHO N arrow Road to a Far Province. Translated by Dorothy Britton. Kodanshu Publ., Tokyo, 1974. ADDENDUM The Pages That Couldn't Wait

I can't deprive my readers (if I still have one at page 509), of the following past-last-minute information about k-tuples of primes. (*) 2-tuples are twin primes, already discussed in Chapter 4, Section III. 3-tuples are triples of primes of the form p, p + 2, p + 6, or p, p + 4, p + 6; it is clear that p, p + 2, p + 4 cannot be a triple of primes. 4-tuples of primes are quadruplets of the form p, p + 2, p + 6, p + 8. More generally, a k-tuple is a sequence of k consecutive primes qi < q2 < ... < qk with qk - qi as small as possible. In other words, qk - q 1 is the smallest integer s with the following prop­ erty; there exist integers bi < b2 < ... < bk , with ~ - bi = s, and for every prime p, not all residues modulo p are represented by some bi (1 < i ~ k). Let n3(x) denote the number of triples of primes (p, p + 2, p + 6) where p ~ x, let

B3 = L (~+ _1_ +_1_) p p+2 p+6 (summation extended over all triples of the form indicated; note that it is necessarily finite). Similarly, let n;(x) denote the number of triples of primes p, p + 4, p + 6, with p ~ x, and

B3=L.,., ,,(1 -+--+-­ 1 1) P p+4 p+6 the corresponding sum.

509 510 Addendum

Again, let 1t4 (x) be the number of quadruples p, p + 2, p + 6, p + 8 with p :::; x and

B4 = I(~ + _1_ + _1_ + _1_) P p+2 p+6 p+8 the corresponding sum. Nicely communicated by letter of October, 1995:

1t3(1.37 x 1014) = 12552824328, 1t~(1.37 x 1014) = 12552911657, 1t4(1.37 x 1014) = 580652391, while B3 = 1.09785103 ... , B; = 0.83711321. .. , B4 = 0.87058837 .... Explicit examples of large k-tuples of primes were communi­ cated to me by T. Forbes:

• Triples (claimed to be the largest known), with 1041 digits: p, p + 2, p + 6, where p = 23456 + 5661177712051. • Quadruples, with 318 digits, p, p + 2, p + 6, p + 8, where p = 21056 + 1301655396715. • 5-tuple, with 174 digits, p, p + 2, p + 6, p + 8, p + 12, where p = 2576 + 39471451430035. • 6-tuple, with 133 digits, p, p + 4, p + 6, p + 10, p + 12, p + 16, where

p = 3 X 10132 + 75543532187. • 7-tuple, with 54 digits, p, p + 2, p + 6, p + 8, p + 12, p + 18, p + 20, where

p = 15 X 1052 + 8496272339051. Addendum 511

• 8-tuple, with 54 digits, p, p + 2, p + 6, p + 12, p + 14, p + 20, p + 24, p + 26, where

p = 15 X 1052 + 11527947572567.

• 9-tuple, with 37 digits, p, p + 2, p + 6, p + 8, p + 12, p + 18, p + 20, p + 26, p + 30, where

p = 22 X 1035 + 111245519978161.

• 10-tuple, with 34 digits, p, p + 2, p + 6, p + 8, p + 12, p + 18, p + 20, p + 26, p + 30, p + 32, where

p = 26 X 1032 + 30697502738421.

• 11-tuple, with 21 digits, p, p + 4, p + 6, p + 10, p + 16, p + 18, p + 24, p + 28, p + 30, p + 34, p + 36, where

p= 300004226699607434083.

• 12-tuple, with 21 digits, p, p + 6, p + 10, p + 12, p + 16, p + 22, p + 24, p + 30, p + 34, p + 36, p + 40, p + 42, where p = 300006711401831244037. • 13-tuple, with 17 digits, p, p + 2, p + 8, p + 14, p + 18, p + 20, p + 24, p + 30, p + 32, p + 38, p + 42, p + 44, p + 48, where

p=28561589689237439.

• 14-tuple, with 17 digits, p, p + 2, p + 8, p + 14, p + 18, p + 20, p + 24, p + 30, p + 32, p + 38, p + 42, p + 44, p + 48, p + 50, where

p= 79287805466244209.

It is easy to conjecture that for every k ~ 2 there exist (like for k = 2), infinitely many k-tuples of primes. But try to prove that there exists one k-tuple. If you do, let me know. Primes up to 10,000

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 6'91 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683

513 514 Primes up to 10,000

2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 6949 6959 Primes up to 10,000 515

6961 6967 6971 6977 6983 6991 6997 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081 8087 8089 8093 8101 8111 8117 8123 8147 8161 8167 8171 8179 8191 8209 8219 8221 8231 8233 8237 8243 8263 8269 8273 8287 8291 8293 8297 8311 8317 8329 8353 8363 8369 8377 8387 8389 8419 8423 8429 8431 8443 8447 8461 8467 8501 8513 8521 8527 8537 8539 8543 8563 8573 8581 8597 8599 8609 8623 8627 8629 8641 8647 8663 8669 8677 8681 8689 8693 8699 8707 8713 8719 8731 8737 8741 8747 8753 8761 8779 8783 8803 8807 8819 8821 8831 8837 8839 8849 8861 8863 8867 8887 8893 8923 8929 8933 8941 8951 8963 8969 8971 8999 9001 9007 9011 9013 9029 9041 9043 9049 9059 9067 9091 9103 9109 9127 9133 9137 9151 9157 9161 9173 9181 9187 9199 9203 9209 9221 9227 9239 9241 9257 9277 9281 9283 9293 9311 9319 9323 9337 9341 9343 9349 9371 9377 9391 9397 9403 9413 9419 9421 9431 9433 9437 9439 9461 9463 9467 9473 9479 9491 9497 9511 9521 9533 9539 9547 9551 9587 9601 9613 9619 9623 9629 9631 9643 9649 9661 9677 9679 9689 9697 9719 9721 9733 9739 9743 9749 9767 9769 9781 9787 9791 9803 9811 9817 9829 9833 9839 9851 9857 9859 9871 9883 9887 9901 9907 9923 9929 9931 9941 9949 9967 9973 10007 10009 10037 10039 10061 10067 10069 10079 10091 10093 TABLE 54. Index of tables.

1. Smallest prime factor In+ I of TI7 =I I; + 1, II = 2. 13 2. Greatestprimefactorqn+lofTI7=lq; + 1, ql = 2. 14 3. Smallestprimefactorrn+l ofTI7=lr; - 1, r l = 3. 15 4. Greatest prime factor Sn+1 of TI7=1 S; - 1, SI = 3. 16 5. Smallest primitive root modulo p. 25 6. Multifactorial primes. 27 7. Valence function of Euler's function. 39 8. Smallest values of Euler's function with given mUltiplicity. 41 9. Fibonacci and Lucas numbers. 70 10. Numbers 2n - 1 and 2n + 1. 71 11. Pell numbers and companion Pell numbers. 72 12. Un(4, 3) and Vn(4, 3). 73 13. Status of Fermat numbers F n , n ::5 22. 87 14. Primes p such that Mp is a Mersenne prime. 94 15. Smallest pseudoprimes to given bases. 110 16. Strong pseudoprimes in bases 2, 3, 5. 116 17. Large Carmichael numbers. 122 18. Number of digits of the largest known Carmichael number with a given number of prime factors. 123 19. Congruences for elliptic pseudoprimes. 134 20. The largest known primes. 158 21. Largest known primes with initial digit d and other digits equal to 9. 162 22. Remarkable factorizations of large numbers. 170 23. Prime-representing polynomials. 192 24. Polynomials representing Fibonacci numbers, Mersenne primes, even perfect numbers, and Fermat primes. 193 25. Large prime triplets in arithmetic progression. 198 26. Comparison of 1r(x) with x/log x, Li(x), and R(x). 237 27. Values of 1r(x). 238

517 518 Index of Tables

28. Roots of the Riemann zeta function. 242 29. Values of '1r2(X) and number of twin primes up to x. 262 30. Largest known twin primes. 264 31. Values of Linnik's constant. 279 32. Strings of primes in arithmetic progression. 287 33. Waring's problem for cubes. 304 34. Waring's problem for powers of 4. 304 35. Waring's problem for powers of 5. 305 36. Waring's problem for powers of 6. 306 37. Waring's problem for powers of 7. 306 38. Waring's problem for powers of 8. 307 39. Waring's problem for powers of 9. 307 40. Waring's problem for powers of 10. 308 41. Bounds for pseudoprimes. 313 42. Count of pseudoprimes. 315 43. Count of Carmichael numbers with a given number of prime factors. 316 44. Count of strong elliptic pseudoprimes. 319 45. Solutions of aP- 1 == 1 (mod p2) for 2 :5 a :5 99 and 3 :5 P < 232. 347 46. Solutions of aP- 1 == 1 (mod p2) for 101 :5 a < 1000 andp < 104 • 348 47. Primes of the form (a" - 1)/(a - 1), a =1= 2, 10. 352 48. Record leading and ending rep digit primes. 354 49. Count of Sierpinski numbers and Riesel numbers. 359 50. Smallest base values yielding generalized Fermat primes. 359 51. Large strings of composite initial values of polynomials. 366 52. Maximum of strings of composite values of binomials X d + k, 2 :5 d :5 5, k with given bounds. 401 53. Count of primes of the form X2 + 1. 403 54. Index of tables. 517 Index of Names

Aaltonen, M., 346, 379, 502 Bachet, e.G., 309 Abel, N.H., 219, 228, 334 Bachmann, P., 495 Adams, W.W., 135,434,451 Backlund, R., 241, 242 Adleman, L.M., 138, 145, 146, Baer, W.S., 305, 306 152,153,171,173,178,330, Baillie, R., 87,110,.111,128,129, 338,342,400,402,449,452, 130, 141, 142, 168,317,449 455,459,460,500,506 Baker, A., 45, 46, 68, 100, 196, Aigner, A., 38, 358, 448, 496 210,212,463,464,465 Aigner, M., 451 Baker, e.L., 235, 477 Albert, A.A., 437, 472, 473 Baker, R., 254, 495 Alcuin of York, 103 Balasubramanian, R., 241, 242, Alford, W.R., 120,315 302,305,308,486,489,490, Amitsur, S.A., 229, 230, 476 491 Anakht, D., 103 Bang, A.S., 43, 67, 338, 437 Angell, 1.0., 351,498 Barban, M.B., 297 Ankeny, N.C., 388, 505 Basho, v, 431, 432, 507 Apostol, T.M., 219, 231, 320, Bateman, P.T., 37,273,322, 483,484 357,399,409,411,412,429, Archibald, R.C., 167,438 443,457,480,482,505,507 Arno, S., 135,458 Bayer, P., 328,499 Artin, E., 43, 262, 295, 379, Bays, C., 235, 275, 485, 486 441,474 Bedocchi, E., 29, 454 Atkin, A.O.L., 121, 148, 149, Beeger, N.G.W.H., 107, 120, 151, 168, 170, 172,264,455, 334,440 459,486 Behara, M., 461 Auric, A., 8, 9 Beiler, A.H., 233, 234, 480 Ayoub, R.G., 200, 219, 231, Bellman, R., 17,2:17,435 433,466,479,483 Bernardi, D., 148 Bernhard, H.A., 1[00,439 Bernstein, D., 172 Bach,E., 145, 166,454,457 Bertrand, J., 247, 397

519 520 Index of Names

Beurling, A., 430 444,445,446,447,452,456, Bichstein, J., 495 482,496,497,501 Bicknell-Johnson, M., Britton, D., 507 450 Brocard, H., 248 Biermann, K.-R., 444 Brown, J., 157, 158,331,356 Bilharz, H., 385, 504 Brun, V., 261, 264, 265, 278, Binder, C., 233 296,331,469 Birch, B.J., 465, 497 Buchstab, A.A., 297, 388 Birkhoff, G.D., 43, 437 Buck, R.C., 187,463 Blundon, W.J., 235, 481 Buell, D.A., 87, 158,356,357, Boender, H., 170 456,501 Bohman, J., 304, 487 Buhler, J.P., 13,26, 171,326, Bombieri, E., 246, 254, 261, 435,502,503 262,278,281,296,297,480, Burgess, D.A., 24, 142,443 490,491,492 Burkhardt, J.C., 233 Bond, R., 350 Buxton, M., 100,447 Borevich, Z.I., 211, 463 Borho, W., 485 Borning, A., 13,26,435 Caldwell, C., 16,26,27, 122, Borodzkin, K.G., 292, 477 123, 157, 159,263,432,436, Borucki, L.J., 355,498 461 Borwein, D., 29, 460 Cantor, M., 284 Borwein, J.M., 29, 460 Capelli, A., 390 Borwein, P.B., 29, 460 Carmichael, R.D., 36, 39, 43, Bosma, W., 148, 150,454 54,63,67,68,69, 118, 119, Boston, N., 204, 205, 467 122,268,393,437,438 Bouniakowsky, V., 372, 386, Carvalho, J.B., 86,461 389,391,400,504 Catalan, E., 98 Bourbaki, N., 147,450,466 Cataldi, P .A., 94 Brancker, T., 233 Cauchy, A.L., 309,467 Brauer, A., 100,236,275,362, Cecuro, E.F., 296 439,474,485,496 Chandrasekharan, K., 462, 464, Bray, H., 344, 497 505 Brent, R.P., 85, 87,100,235,241, Chaum, D., 453 252,261,262,450,451,457, Chein, E.Z., 99, 449 459,483,484,487 Chen, J.R., 246, 264, 279, 292, Bressoud, D.M., 164,457 295,299,302,305,332,477, Breusch, R., 229, 397, 476 479,481,483,486,487,488,492 Brillhart, J., 19,50,51,53,82, Chernac, L., 233 85,87,97, 151, 164, 165, 167, Chernick, J., 120, 121, 122, 132, 169,272,345,351,363,432, 379,438 Index of Names 521

Chowla, S., 200, 206, 208, 277, Dase, Z., 233 279,281,285,465,466,471, Davenport, H., 254, 261, 305, 474 306,433,473,474,480 Chudnovsky, D.V., 87,152,454 Davies, R.O., 250, 399 Chudnovsky, G.V., 87,152,454 Davis, J.A., 165, 168,351,452, Cipolla, M., 106, 107, 108, 109, 500 117,437 Davis, M., 190, 196,464 Clark, D., 334 Dedekind, R., 209, 324 Clarke, A.A., 436 de Koninck, J.M., 493 Clarkson, J.A., 217 Delange, H., 484, 485 Clausen, T., 420 De la Vallee Poussin, C.J., 215, Clement, P .A., 259, 260, 475 225,226,229,243,274,429 Coblyn, 260, 469 Deleglise, M., 236, 237, 494 Cohen, G.L., 37, 100, 101,449, De Leon, M.J., 498 456,457,459 Delmer, F., 303, 493 Cohen, H., 147,231,453,456 Denes, P., 329,496 Cohn, N., 200, 463 Deshouillers, J.M., 290, 296, Cole, F.N., 163, 167 298,303,305,482,484,491, Colquitt, W.N., 94, 158,458 493,494 Confucius, 104 Deuring, M., 210, 211, 462 Conrey, J.B., 241, 488, 490, 492 De Weger, B.M.M., 68, 457 Cook, E.J., 307, 308 Diamond, H.G., 229, 488 Coppersmith, D., 337, 502 Diaz, J .B., 355, 498 Costa Pereira, N., 237, 239, Dickson, L.E., 8, 90, 98, 99, 490,491 104,167,199,233,284,300, Couvreur, C., 451 303,304,305,372,373,391, Cox, C.D., 12,435 437,472,473,504 Cramer, H., 253, 371,472 Dieudonne, J., 485 Crandall, R.E., 13,26,86,87, Diffie, W., 173,447 326,334,434,435,461,502, Dilcher, K., 334, 503 503 Diophantus, 309 Crelle, A.L., 233 Dirichlet, G.L., 211, 265, 273, Crocker, R., 109,443 274,461,468 Chda Kov, N.G., 471, 473 Dixon, J.D., 184, 191,450,453 Cullen, J., 360 Dodson, B., 170 Cunningham, A.J.C., 87, 151, Doenias, J., 86, 87,461 164,167,169,360,438 Dress, F., 305, 491, 493, 494 Dressler, R.E., 322,482 Du, S., 105,456 Daboussi, H., 231, 489 Dubner, H., 12, 13, 16,26,86, Damgard, 1.,494 89,96, 121, 122, 123, 124, 522 Index of Names

Dubner, H. (Cont.) Euler, J.A., 301, 304, 305, 306, 125, 158, 160, 161, 162, 169, 307,308 198,263,264,330,331,333, Euler, L., 6, 8,21,22,34,36, 350,351,353,354,355,356, 46,47,48,54,84,87,90,94, 359,360,363,414,432,435, 99, 100, 102, 130, 167, 199, 436,454,457,461,466,500, 204,215,216,217,218,219, 501,502,503 229,291,329,378,409 Dubner, R., 454 Dubuque, W., 147 Dudley, U., 186 Fauquembergue, E., 94 Duparc, H.J.A., 111, 120,440 Fee, G., 329, 502 Durst, L.K., 74, 442 Feinstein, A., 480 Dux, E., 217 FeIgner, U., 237, 249, 493 Felkel, A., 233 Feller, W., 232, 477 Eberhart, 412, 413 Felthi, C., 206, 207, 465 Edwards, A.W.F., 17,435 Fermat, P., 46, 48,51,54,55, Edwards, H.M., 225, 242, 483 83,90,104,300,308,309, Eisenstein, F.G., 88, 335,436 340, 342, 343 Elliott, P.D.T.A., 142,278,321, Ferrero, B., 328, 499 445,482,486 Fibonacci, 53 Ellison, F., 485 Finn, M.V., 298, 491 Ellison, W.J., 289, 303, 311, Fjellstedt, L., 272, 476 434,482,485 Flath, D.E., 208, 467 Elmore, S., 100,447 FleCK, A., 304, 305, 306 Eratosthenes, 20, 177 Forrest, J., 299, 487 Erdos, P., 40, 45, 46, 104, 107, Forti, M., 243 109,111,142,215,217,227, Fouvry, E., 262, 312, 330, 338, 228,237,247,254,255,256, 342,488,490,500 265,278,280,281,282,286, Frasnay, C., 32, 458 311,312,313,317,321,322, Fridlender, V.R., 24, 439 332,340,342,357,430,438, Friedlander, J.B., 206, 262, 278, 439,442,443,444,447,455, 465,491 470,471,472,473,474,475, Fritsch, R., 461 477,479,487,492,496,498, Frobenius, F.G., 199,200,201, 499 336,462 Ernvall, R., 182, 326,465, 503 Froberg, C.E., 304, 487 Escott, E.B., 199 Frohliger, J.A., 298, 491 Estermann, T., 268, 292, 297, Fung, G.W., 502 298,472,479 Furstenberg, H., 10, 11,435 Euclid, 3, 11,99 Furtwangler, P., 495 Index of Names 523

Gage, P., 94, 95 Granville, A., 120,283, 298, Gagola, G.M., Jr., 453 299,315,329,336,337,340, Gandhi, J.M., 182, 183, 185, 342,345,379,414,432,460, 464 492,493,494,500,501,502 Gardner, M., 171, 198,448,466 Greenleaf, N., 463 Gauss, C.F., 19,23,24,46,47, Greenwood, M.L., 205, 467 88,138,202,208,209,210, Grolz, W., 272, 481 211,212,215,220,221,233, Gross, B., 212, 466 239,309,436 Grosswald, E., 24, 47, 285, 286, Gautschi, N., 494 444,450,486,488,505 Gelfond, A.O., 210, 293, 480 Gruenberger, F.J., 235, 477 Gerig, S., 227, 484 Griin, 0., 101,440 Germain, S., 91,329 Guillaume, D., 121,459 Gerst, I., 272, 482 Gunderson, N.G., 336, 337,495 Gerver, J.L., 165,452 Gupta, H., 38, 440 Gilbreath, N.L., 257 Gupta, R., 385, 386, 489, 506 Gillies, D.B., 94, 412, 444, Gurak, S., 135, 458 505 Gurgensohn, R., 29, 460 Giordano, G., 253, 492 Guy, L., 465 Giuga, G., 28, 446 Guy, R.K., 12, 17,99, 103, 165, Godwin, H.J., 351, 498 434,435,447 Goetgheluck, P., 206, 467 Goldbach, C., 291, 292, 294, 295,296,297,299,311,404 Hadamard, J., 215, 225, 226, Goldberg, K., 346, 496 229,231,429 Goldfeld, D., 212,465, 466 Hagis, P., Jr., 37, 99, 100, 101, Goldstein, L.J., 434 286,445,446,449,452,486 Goldwasser, S., 148, 155,455 Halberstam, H., 259, 261, 278, Golomb, S.W., 11, 184,340, 281,283,297,313,373,388, 435,464,497 391,434,481,482,483,498 Golubev, A. V., 287 Hall, M., Jr., 366, 495 Gonnet, G., 336 Hammond, N., 299, 487 Gonter, R.H., 350, 502 Han, Q., 104 Goodstein, R.L., 463 Hardy, G.H., 29, 91, 179,205, Gordon, D.M., 133, 134,317, 215,240,242,246,248,262, 318,319,457,492,493 265,285,289,292,294,297, Gordon, J., 157 298,301,302,303,305,306, Gostin, G., 461, 467 307,308,320,371,377,390, Graham, R.L., 367, 497 396,403,408,409,430,433, Graham, S., 279, 315, 488, 495 469,470,473,504 Gram, J.P., 225, 229, 241, 242 Harman, G., 254, 495 524 Index of Names

Haselgrove, C.B., 241, 242 Hudson, R.H., 235, 275, 486, Hasse, H., 150,362,475,497, 490 504 Huenemann, J., 121, 122,452 Hatcher, W.S., 195,466 Huizing, 1., 171 Hausman, M., 42, 449 Hundsam, 234 Haworth, G., 98 Hurwitz, A., 4,5,86,87,94, Heath-Brown, D.R., 100,241, 307,434,443,444 254,280,281,284,285,330, Hutchinson, J.1., 241, 242 332,338,342,385,386,486, Huxley, M.N., 243, 254, 483, 488,490,493,494,500,506 485 Heegner, K., 210, 463 Heieh, S.K., 480 Heilbronn, H., 210, 211, 215, I, see Ribenboim, P. 302,387,462,472,473, Indlekofer, K.-H., 495 504 Ingham, A.E., 243, 254,313, Hellman, M.E., 173,447 399,471,472,473 Hendy, M.D., 200, 201, 464 Inkeri, K., 83, 346, 350,443, Henry, C., 90 502 Hensley, D., 248, 483 Ishikawa, H., 247, 249, 471 Hilbert, D., 301, 302, 468 Israel, R.B., 378, 506 Hildebrand, A., 256, 332, 492 Ivic, A., 225, 242, 243, 246, 253, Hlawka, E., 233, 288, 486 487,489,490 Hodgson, B.R., 195,466 Iwaniec, H., 246, 254, 262, 278, Hoggatt, V.E., 450 280,313,405,486,488,489, Hoheisel, G., 253, 471 492, 506 Holdridge, D.B., 165, 168,351, Iwasawa, K., 327, 328,497 452, 500 Honsberger, R., 104, 446 Hoogendoorn, P.J., 451 Jacobi,C.G.,47,48, 85, 335 Hooley, C., 361, 383, 385, 498, Jacobsthal, E., 280, 281, 505 478 Horadam, A.F., 455 Jaeschke, G., 115, 116, 123, Horn, R.A., 399,409,411,412, 316,357,460,493,500 505 Jantzen, J.C., 485 Hornfeck, B., 102, 272, 441, Jarai, A., 495 442, 482 Jarden, D., 65, 169,252,363, Hua, H.F., 105 442,481 Hua, L.K., 247, 310, 434, 477 Jeans, J.H., 104 Huang, M.D.A., 152, 153,455, Jensen, K.L., 325,415 459 Joffe, S.A., 234, 474 Huddleston, H., 171 Johnson, W.,326,498 Index of Names 525

Jones, J.P., 29,155,191,192, Konheim, A.G., 178,450 193,194,195,465,466 Korobov, N.~., 477 Jones, ~.F.,235,481 Korselt, A., 118,437 Jungnickel, D., 451 Kraft, H., 485 Jurkat, VV.B., 231,297,478 Krasner, ~., 329,495 Jutila, ~., 243, 254, 279, 485, Kravitz, S., 412, 505 486 Kronecker, L., 209,289 Kruger, J.G., 233 Kruyswijk, D., 345, 497 Kac, ~., 321, 478 Kubina, J.~., 303,493 Kamke, E., 309, 311, 470 Kuhn, P., 405,504 Kanold, H.J., 43, 99, 100, 101, Kuhnel, U., 99, 100,440 102,280,281,399,438,439, Kulik, J.P., 233, 234 440,441,442,479 Kumar ~urty, V., 281, 485 Kapferer, H., 275, 479 Kummer, E.E., 4, 31, 32, 195, Kaplansky, I., 91,191,439 209,323,324,325,327,328, Karacuba, A.A., 242 415,434,436,462 Karst, E., 204, 464 Kundert, E.G., 350, 502 Kearnes, K., 24, 453 Kusjasev, A.A., 296 Keller, VV., 86, 89, 97, 98,123, Kutrib, ~., 495 159,168,234,264,331,346, Kutsuna, ~., 207, 466 350,356,357,358,359,360, 361,363,452,454,459,461, 489,500,502,503 Laborde,P.,102,441 Kempner, A.J., 304, 306 Lagarias, J.C., 236, 362, 490, Kendall, D.G., 103,451 501 Khinchin, A.Ya., 295, 474 Lagrange, J.L., 57,90,188, Kilian, J., 148, 155,455 208,285,294,300,301,309, Killgrove, R.B., 257, 478 378 Kishore, ~., 99,100,101,448, Lal, ~., 215, 481 450, 453 Lambert, J .H., 233 Kiss, P., 65,112,118,131,317, Lame, G., 137 449,455,456,492 Landau, E., 24, 211, 215, 221, Klee, V.L., 39, 40, 439 226,231,244,268,282,293, Klimov, N.I., 296 304,320,337,429,433,462, Kloss, K.E., 345, 350, 497 468, 471, 472 Knodel, VV., 125,440 Lander, L.J., 287, 481 Knuth, D.E., 450 Landrock, P., 494 Koblitz, N., 178,456 Landry, F., 85, 87, 167 Kolesnik, G.A., 246, 290, 481 Lang, S., 441, 474 Konhauser, J., 351, 503 Langevin, ~., 237, 485 526 Index of Names

Leavitt, W.G., 260, 488 Lioen, W., 171 Leech, J., 275, 477 Liouville, J., 304, 323 Legendre, A.M., 30, 31, 32, 43, Littlewood, J.E., 205, 239, 242, 46,215,219,220,233,267, 245,246,248,262,265,275, 268,272,329 285,292,294,297,298,302, Lehman, R.S., 239, 481 303,305,306,307,308,371, Lehmer, D.H., 36, 37, 48, 50, 377,390,396,403,408,409, 53,69, 74, 82, 83, 88,91,93, 430,469,470,504 97, 106, 107, 113, 114, 118, Liu, J., 279 119, 127, 130, 137, 145, 164, Loh, G., 121, 122, 198,333, 167,168,169,200,233,234, 461,502 236,241,242,311,334,438, Lou, S., 254, 262, 494 439,445,446,447,452,462, Louboutin, S., 203, 467 463,478,499 Low, M.E., 273, 480 Lehmer, D.N., 83,234,469 Lucas, E., 32,48,49,50,54,55, Lehmer, E., 93, 118,444 56,64,65,74,84,85,87,90, Leibniz, G.W., 104 91,93,94, 117, 135, 154,304, Lemos, M., 178, 457 436,437,495 Lenstra, A.K., 86, 87, 165, 166, Liineburg, H., 43, 451 170,171,432,456,458,460 Lenstra, H.W., Jr., 86, 145, 147, 148, 149, 163, 165, 166, MaaB, H., 462, 464, 505 412,449,450,451,453,455, MacLaurin, C., 219, 222, 228 456,458,460,506 Mahler, K., 303, 477 Le Veque, W.J., 445 Maier, H., 254, 255, 256, 257, Levesque, C., 493 488,490,492,493 Levinson, N., 241, 483 Maillet, E., 301, 302, 304, 305, Levit, R.J., 100,439 306,307,308 Li, S.L., 104, 105 M~kowski, A., 125,311,312, Li, Y., 105,456 443,483 Lienard, R., 385, 504 MaIm, D.E.G., 114, 130,448 Lieuwens, E., 37, 109, 112, 129, Malo, E., 106, 437 131,445,455 Manasse, M.S., 86, 87, 165, Ligh, S., 90, 446 166,170,172,458,460 Light, W.A., 299, 487 Mann, H.B., 29, 295, 445, 474 LindelOf, E., 246 Masai, P., 39,451 Linfoot, E.H., 210, 462 Masley, J.M., 327,497,498 Linnik, Yu.V., 24, 210, 277, Masser, D.W., 448 279,280,281,293,303,304, Matijasevic, Yu.V., 155, 190, 310, 474, 480 191,192,194,195,464,465 Lintz, R.G., 461 Mattics, L.E., 339, 500 Index of Names 527

Maurer, U.M., 178,458 Montgomery, P.L.., 165, 169, McCarthy, D., 438, 439, 447, 170,171,432 478 Montgomery, P.M., 171 McCarthy, P.J., 100,442 Morain, F., 85, 87,97,121,148, McCurley, K.S., 138,400,401, 149,151,168,353,369,432, 402,403,455,460,501,506, 456,458,459 507 Moran, A., 286, 495 McDaniel, W.L., 100, 117, 118, Mordell, L.J., 133,211,462 445,446,456 Morimoto, M., 359, 501 Me, see Ribenboim, P. Morrison, M.A., 79,85,87, Meissel, E.D.F., 178, 181,215, 165,169,445,447 235,236,275,468 Morrow, D.C., 126,440 Meissner, W., 334 Moser, L., 217, 242, 247, 475 Mendelsohn, N.S., 38, 358, 447, Motohashi, Y., 278, 482 498 Mozzochi, C.l., 33, 254,302, Mendes-France, M., 289, 434 308,444,489,491 Mersenne, M., 90, 93, 95, 168, Mullin, A.A., 11,260,435,488 355, 356 Murata, L., 386,414,507 Mertens, F., 215, 222, 231 Muskat, J.B., 101,444 Metrod, G., 8, 9 My, see Ribenboim, P. MetsankyHi, T., 326, 328, 329, 498, 503 Miller, G.L., 144, 145, 157,447 Nagell, T., 268, 272,387,470, Miller, V.S., 236, 490 476,481,504 Mills, W.H., 101, 102, 186,446, Nagura, l., 252, 476 463 Nair, M., 237, 488 Min, S.H., 246 Narasimhamurti, V., 307, 308, Mimic, J., 181 473 Mirimanoff, D., 334, 336 Narkiewicz, W., 208, 298, 464, Mirsky, L., 332, 496 494 Miyamoto, J., 318,493 Naur, T., 12, 167, 169,451, Mobius, A., 229 453 Mohanty, S.P., 17,435 Neal, L., 446 Mollin, R.A., 202, 203, 206, Needham, J., 104 207,208,340,341,432,466, Nelson, H., 94, 158,457 467,501 Neubauer, G., 231, 479 Monagan, M.B., 336,414, 502 Newman, D.J., 226, 303, 478, Monier, L., 110, 113, 115,450 487 Montgomery, H.L., 212, 241, Newman, M., 367,499 243,254,299,327,346,363, Newton, I., 384 464,481,482,483,484,498 Ng, H.H., 477 528 Index of Names

Nicely, T.R., 495 Pascal, M., 140 Nichomachus of Gerasa, 103 Patashnik, 0., 432 Nickel, L., 94, 450 Patterson, S.J., 492 Nicolas, J.L., 147,320,322, Peano, G., 180,400 453,489 Pearson, E.H., 350, 496 Niebuhr, W., 121, 122,461, Pell, J., 54 503 Penk, M.A., 13,26, 168, Niven, 1., 268, 277, 288, 303, 435 474,477,484 Pepin, T., 51, 84, 154,436 Noll, L.C., 94, 157, 158,331, Perisastri, M., 101,442 356,450 Perott, J., 8 Norrie, C., 86, 87,461 Perrin, R., 135,437 Norton, K.K., 101,443 Pervushin, 1.M., 87, 93, 94 Nowakowski, R., 12, 17,435 Phong, B.M., 65, 112, 118, 131, 449,455,456 Pillai, S.S., 24, 303, 305, 438, Obhith, R., 350,495 439,471,472,473 Odlyzko, A.M., 231, 236, 241, Piltz, A., 253 242,258,357,400,401,402, Pinch, R.G.E., 123,316,460, 490,492,494,499,506 494 Odoni, R.W.K., 18,435 Pintz, J., 153,220,231,254,299, Oesterle, J., 212, 466 313,457,487,488,489,492 Ondrejka, R., 157,454 Pirandello, L., 140,437 Onishi, H., 388, 505 Pisot, c., 484 Opperman, L., 248, 257 Pla.isted, D.A., 155, 157,449 Ore, 0., 37,101,102,439,441, Pocklington, H.C., 48, 51, 52, 444 53, 79, 150, 156, 157,437 Orzech, M., ix Poitou, G., 484 Ostmann, H.H., 311, 477 Polignac, A. de, 250, 252, 374 Pollaczek, F., 336, 337, 495 Pollard, J.M., 85, 87, 165, 168, Pan, C.B., 297 446,447,450,458,460 Pan, C.D., 279, 297, 299, 480, P6lya, G., 4, 435 487 Pomerance, C., 37,40,99, 100, Pandora, 148 101, 102, 110, 111, 114, 115, Pappalardi, F., 386, 507 116, 120, 145, 146, 154, 155, Parady, B., 157, 158,263,331, 165, 166, 167,249,255,281, 356,493 283,298,312,313,314,315, Parberry, E.A., 127, 128, 130, 317,318,321,332,334,351, 131,445 379,400,432,446,447,448, Parkin, T.R., 287, 481 450,451,452,453,455,456, Index of Names 529

460,486,487,488,492,493, Reiner, I., 187, 463 494,498,503 Renyi, A., 264, 297, 474 PotIer, A., 251, 493 Reuschle, C.G., 167 Poulet, P., 106, 122, 167,315, Ribenboim, P., x, 47, 69,82,95, 473 102, 116, 127, 135, 136, 138, Powell, B., 24, 250, 255, 256, 140, 143, 144, 148, 156, 160, 268,270,277,332,339,342, 161,166,169,171,175,177, 345,378,453,484,485,487, 182,188,196,199,200,205, 489,490,498,499,500,506 206,208,209,212,213,214, Powers, R.E., 94 215,220,221,223,225,229, Prachar, K., 255, 274, 282, 400, 233,235,239,243,245,253, 433,478 254,255,258,262,269,270, Pratt, V.R., 154, 155,447 272,276,277,279,281,292, Pritchard, P .A., 286, 287, 488, 293,296,297,299,300,301, 490,495 310,311,317,323,324,327, Pritchard, R.E., 432 328,329,331,335,336,338, Proth, F., 48, 52,257,437,468 340,342,343,344,345,346, Puccioni, S., 338, 497 355,356,361,362,363,364, Putnam, H., 189, 190,463 365,367,371,372,373,374, Pyatetskii-Shapiro, LL, 290, 476 375,376,377,378,379,380, 381,382,383,385,386,387, 388,389,392,393,396,399, Quisquater, J. J ., 451 400,403,404,413,414,415, 417,422,425,427,428,429, 430,431,432,434,446,461, Rabin, M.O., 114, 123, 144, 464,466,467,499,500,501, 146, 157,447,450 502, 506 Rabinovitch, G., 199,200,202, Ricci, G., 256, 297,388,391, 462 398,405,476 Rademacher, H., 264, 297, 420, Richards, L, 248, 483 470,506 Richert, H.E., 244, 259, 261, Rado, R., 482 281,283,284,296,297, Ralston, K.E., 257, 478 373,388,391,434,475,481, Ramachandra, K., 243 483 Ramanujan, S., 247,469 Rickert, N.W., 170, 172,264, Raman!, 0., 296, 495 486,498 Ram Murty, P.M., 273, 318, Rieger, G.J., 302, 303, 389, 476, 385,386,493,506,507 505 Rankin, R.A., 255, 402, 473, 479 Riemann, B., 215, 219, 221, Realis, S., 304 222,223,224,225,239,240, Recaman Santos, B., 42, 448 244,429 530 Index of Names

Riesel, H., 83, 87,94,97,164, Schatunowsky, J., 281 296,345,357,359,360,454, Scherk, P., 295, 474 489,496 Schinzel, A., 37, 38, 39,42,44, Rivat, J., 236, 237, 494 45,68,69, 74, 88, 109, 119, Rivest, R.L., 171, 173, 176, 178, 128,275,276,280,282,294, 449 296,372,373,377,389,390, Robbins, N., 99, 446 391,393,394,396,399,400, Robin, G., 249, 489 404,409,441,442,443,444, Robinson, J., 190, 195 445,446,477,478,479,504, Robinson, R.M., 93, 94, 97, 505 360,441,442 Schlafly, A., 40 Robson, R., 171 Schnirelman, G., 293, 294, 295, Roe, S., 299, 487 296,303,311,471 Rohlfs, J., 485 Schoenberg, I.J., 320, 470 Rohrbach, H., 252,397,480 Schoenfeld, L., 233, 238, 239, Romani, F., 304, 488 241,245,248,253,320,479, Ross, P.M., 297, 484 481,482,484 Rosser, J.B., 238, 239, 241, 242, Sch6nhage, A., 137,445,479 245,248,249,320,473,479, Schoof, R.J., 148, 151, 165, 481,484 454 Rotkiewicz, A., 97, 108, 110, Schor, P.W., 166,461 118,131,268,312,313,314, Schorn, P., 5 344,350,442,444,446,453, Schroeder, M.R., 213, 412, 413, 478,479,480,481,483,486, 434,490,506 487,497,501 Schroeppel, R., 116 Rouse Ball, W. W., 90 Schuh, P., 37, 439 Rubugunday, R., 303,474 Schur, 1.,272, 308,469 Ruby, R., 171,203 Seah, E., 353,499 Rumely, R.S., 145, 146,400, Segal, S., 248, 479 452,453 Selberg, A., 215, 227, 228, 229, 231,240,255,274,283,297, 430,474,475,476 Saaty, T.L., 444 Selfridge, J.L., 12,37,38,50, Salie, H., 24, 440 51,53,82,86,87,88,97,110, Sambasiva Rao, K., 306 114, 115, 116, 130, 141, 145, Samuel, P., 9, 435 164, 169,249,312,314,315, Sark6zy, A., 317, 332,492 334,440,444,446,450,452, Sarrus, F., 105 457,487,496 Sasaki, R., 202, 466 Seredinskij, V.N., 287 Sato, D., 155, 191, 192,465 Sergusov, I.S.A., 260, 482 Satyanarayana, M., 101,442 Serre, J.P., 309, 413, 482 Index of Names 531

Seshu Aiyar, P. V., 469 Somayajulu, B.S.K.R., 38,440 Shafarevich, I.R., 211, 463 Somer, L., 117, 131, 132, 454, Shafer, R.E., 256, 489 457 Shallit, J., 166,457 Sompolski, R.W., 326, 502 Shamir, A., 171, 173, 178,449 Sora, 431 Shanks, D., 11,29,98, 112, 135, Specker, E., 4 251,261,275,337,367,389, Spiro, C.A., 332,499,501 401,409,412,433,436,443, Srinivasan, S., 386, 507 445,451,478,484,499,505, Stark, H.~., 210, 212, 463, 464, 506 465 Shapiro, H.N., 42, 99, 228, 296, Steiger, W.L., 153,457 440,449,476 Stein, ~.L., 299, 480 Shen,~.K., 118,299,455,480 Stein, P.R., 299, 480 Shorey, T.N., 46, 74, 447, 451 Stemmler, R.~., 303, 306, 307, Siebert, H., 296 308,480 Siegel, C.L., 196,211,212,372, Stephens, P.J., 498 415,425,462,464,497,505 Stern, ~.A., 407 Sierpinski, W., 37, 38,40,42, Steuerwald, R., 109,439 89,182,249,268,276,280, Stewart, C.L., 45, 46, 68, 69, 355,357,358,362,372,373, 74,447,448,451 377,387,389,391,396,397, Stieltjes, T.J., 4, 9, 231, 434 399,404,409,433,441,442, Stirling, J., 218 463,477,480,496,504,505 Strassen, V., 113, 114, 137, 143, Silverman, J.H., 133,455,461 144,445,448 Silverman, R.D., 165, 169, 170, Straus, E.G., 265, 487 172,363,456,458,460,501 Stridsberg, E., 302, 469 Singmaster, D., 129,453 Suryanarayana, D., 101,444, Sinha, T.N., 103,446 445 Sinisalo, ~.K., 299, 494 Suzuki, J., 337, 503 Siu, ~.K., 104, 105 Swift, J.D., 315,484 Skewes, S., 239,471,476 Swinnerton-Dyer, H.P.F., Skolem, T., 193, 463 465 Skula, L., 329, 499, 501, 503 Sylvester, J.J., 82, 237, 468 Slowinski, D., 93, 94, 95, 96, Szego, G., 4, 435 158,412,413,449,506 Szekeres, G., 201, 465 Small, C., 36, 311, 485 Szemeredi, E., 153,457 Smith, G., 157, 158,331,356 Szymiczek, K., 311, 317, 481 Smith, J., 157, 158,264,331, 356,493 Solovay, R., 113, 114, 143, 144, Tanner, J.W., 326, 337, 501, 448 502 532 Index of Names

Tate, J., 133,461 Valette, A., 39, 451 Taylor, B., 218 Van de Lune, J., 241, 299, 491, Taylor, R., 504 492 Templer, M., 13,435 Vanden Eynden, C., 183,217, Te Riele, H.J.J., 100, 171,231, 464,487 239,241,299,459,490,491, Van der Corput, J.G., 285, 298, 492 472,473 Thanigasalam, K., 305, 306, Van der Poorten, A.J., 12,314, 307,308,310,488,491, 435,487 492 Vandiver, H.S., 43,327,336, The Queen, 135,480 346,414,437 The Three Knights of Numerol- Van Emde Boas, P., 138,452 ogy,116 Van Leeuwen, J., 458 Thomas, H.E., 305,484 Vardi, I., 151 Thomas a Kempis, 427 Vaughan, R.C., 296, 299, 305, Thue, A., 7, 434 306,307,308,332,471,484, Thyssen, A., 286, 495 489,491,498 Tijdeman, R., 451, 452 Velleman, D., 351, 503 Titchmarsh, E.C., 225, 232, Vinogradov, I.M., 24, 281, 289, 241,246,471,476 292,295,297,298,302,310, Tonascia, J., 345, 497 430,471,472,473,475,478 Torelli, G., :15, 468 Vinogradov, J.M., 477 Touchard, J., 101,441 Viola, C., 243 Traub, J.F., 138,448 Von Gumpach, J., 105 Trevisan, V., 86, 461 Von Koch, H., 245, 468 Trost, E., 247,249,433 Von Mangoldt, H., 215, 224, Tschebycheff, P.L., 177,215, 226,229,240,244 220,221,247,252,275,293, Von Neumann, J., 135 397,420,429 Von Staudt, C., 420 Tschudakoff, N.G., 245, 298 Von Sterneck, R.D., 231 Tuckerman, B., 94, 97,100, Vorhoeve, M., 452 164,445,452 Vorob'ev, N.N., 496 Tunin, P., 253, 256, 277, 289, Voutier, P.M., 69, 461 472,475,482,490 Turing, A., 93, 242 Tzanakis, N., 68, 345, 457, Wada, H., 155, 191, 192,465 501 Wade, A., 104 Wade, T., 104 Wagon, S.,40, 138, 145,242, Uchida, K., 327, 497 351,454,455,491,503 Veno Castle, Lord of, 431 Wagstaff, S.S., Jr., 11, 12,97, Index of Names 533

110, 111, 114, 115, 116, 121, Wiens, D., 155, 191, 192,465 122, 128, 129, 130, 141, 142, Wiles, A., 504 164, 165, 167,281,282,283, Willans, C.P., 180, 181, 182, 312,314,315,317,326, 334, 463 337,351,412,425,432,436, Williams, H.C., 83, 85, 89, 114, 449,450,452,453,457,460, 131,132, 145, 165, 169,207, 486,487,499,501,502,506 337,351,353,363,364,367, Wakulicz, A., 42, 443 445,446,449,452,453,460, Walfisz, A.Z., 232, 244, 246, 466,499,500,501,502 479 Williams, S.M., 403 Wall, e.R., 321,483 Wilson, 1.,21,26,48, 180, 181, Wall, D.D., 364,496 346, 350 Wall, D.W., 37,450 Winter, D.T., 171,241,491 Walsh, P.G., 340, 341, 501 Wintner, A., 321, 470 Wang, Y., 279, 292, 295, 297, Wirsing, E., 102,442,443 299,480,490,492 W6jcik, 1.,273,482 Ward, M., 67, 74, 82, 366,441, Wolfskehl, P., 281, 468 443,496 Wolstenholme, 1., 29 Waring, E., 300 Woodall, H.l., 164, 360,438 Warren, L.l., 344,497 Woods, D., 121, 122,452 Wasen, R., 312, 486 Wooldridge, K., 321, 487 Washington, L.C., 9, 10,328, Wormell, C.P., 463 435,499 Wrench, 1.W., 261, 262, 385, Waterhouse, W.C., 436 478,484,505 Watson, G.L., 304,476 Wright, E.M., 29, 91,179,186, Webber, G.C., 100,440 229,289,301,320,433,463,476 Weil, A., 133, 385, 504 Wunderlich, M.e., 148, 169, Weinberger, P.l., 211, 345, 464, 303,405,453,483,493,506 497 Wylie, 105 Weintraub, S., 251, 287, 485, 488,493,494 Weis, 1., 252, 397,480 Yao, A.C., 155,448 Welsch, L., lr., 94, 158,458 Yao, Q., 254, 494 Wendt, E., 268, 468 Yates, S., 157, 159,263,351, Western, A.E., 87, 91, 438 453,454,459,500 Westzynthius, E., 255, 256, 471 Yohe, 1.M., 241, 481 Weyl, H., 289, 469 Yorinaga, M., 121, 122, 127, Wheeler, D.l., 97 128,315,448,449,450 Wieferich, A., 304, 305, 306, Young, 1., 86, 87, 158,251,287, 333,334,336,337,342,469 356,357,456,461,493,501 Wiener, N., 226, 430 Yu, K., 480 534 Index of Names

Zachariou, A., 33 Zarnke, C.R., 89,445 Zagier, D., 212, 239, 466, 485 Zhang,~., 121,459 Zarantonello, S., 157, 158,264, Zsigmondy, K., 43, 67, 437 331,356,493 Subject Index

Abelian varieties, primality tests Carmichael-Lucas numbers, 132 with, 152 Carmichael numbers, 118-126 Additive number theory, 294 distribution of, 314-317 Almost-primes, 258-259 infinitely many, 379 method of, 296 Carmichael's conjecture, 39-40, APR test, 146-147 393 a-pseudoprimes, 108-112 Certificate of p, 154 Arithmetic progression Chen's theorem, 297 prime numbers in, 265-287 Chinese congruence, 103 smallest prime numbers in, Chinese remainder theorem, 33- 277-284 34 strings of primes in, 284-287 Chinese theorem, 104-105 Artin's conjecture, 383-386 Class number problem, 208-212 Artin's constant, 384 Companion Lucas sequences, 55 Auric's proof, 9 Complex multiplication, 133 Complex number, 266 Composite Mersenne numbers, Baillie and Wagstafrs method, 378 141-142 Composite numbers, 1, 162-178 Bang's special case, 43 large, factorization of, 163- Basis method, 293 172 Bernoulli numbers, 217-218 Congruences Bernoulli polynomials, 219 classical primality tests based Bertrand's postulate, 247 on, 48-53 Bibliography, 433-507 fundamental theorems on, 21- Binomials, sequences of, 42-46 48 Bouniakowsky's hypothesis, Consecutive primes, 287 386-387,389 Cryptography, public key, 173- Brun's constant, 261 178 Cubic polynomials, 206 Cullen numbers, 360-361 Capelli's theorem, 394 Cunningham chains, 333

535 536 Subject Index

Curious primes, 159-162 valence of, 38-42 Cyclotomic numbers, 82 values of, 37-38 Cyclotomic polynomials, 267 Euler's proof, 6-7 Even pseudoprimes, 107

De Leon's lemma, 342 Density hypothesis, 243 Factorials, powers of prime Density of primes dividing V, numbers dividing, 30-33 361-362 Factorization(s), 19-20, 165-166 Diophantine polynomial, univer­ of large composite numbers, sal, 194-195 163-172 Diophantine set, 189 nonuniqueness of, 324 prime number set as, 190 remarkable, before comput­ Dirichlet convolution, 230 ers, 166-172 Fermat numbers, 4-5, 83-90 factors of, 355 ECPP (elliptic curve prime prov­ generalized, 89, 359-360 ing), 148-152 status of, 87 Elliptic curve over K, 149 Fermat quotients, 335-336 Elliptic curve prime proving tables of, 345-349 (ECPP), 148-152 Fermat's little theorem, 22-25, Elliptic curves, primality tests 49 with, 148-152 Fibonacci numbers, 53-54 Elliptic pseudoprimes, 132- Lucas numbers and, 70 134 Fibonacci pseudoprimes, 127- distribution of, 318-319 129 Eratosthenes, sieve of, 20-21 Fundamental prime number the­ Erdos's conjecture, 342 orem, 225-227 Euclidean sequences, 11-12 Fundamental theorem in arith­ Euclid's proof, 3-4 metic,I-2 Euler-Lucas pseudoprimes, FUrstenberg's proof, 10-11 130-131 Euler-MacLaurin summation formulas, 219, 222 Gamma function, 223 Euler mUltiplicity of m, 39 General purpose factoring algo­ Euler pseudoprimes, 112-113 rithms, 166 Euler's constant, 222 Generalized Fermat numbers, Euler's function, 34-42 89, 359-360 distribution of values of. 319- General references, 433-434 322 Gigantic primes, 159 growth of, 42-43 Giuga, property of, 28-29 Subject Index 537

Goldbach problem, odd, 292- Legendre symbol, 46-47 293 Lehmer numbers, 69, 74 Goldbach's conjecture, 291-299 Lehmer's function, 36-37 Goldbach statement, odd, 291 L-functions, 274 Goldwasser-Kilian algorithm, Lindelofs hypothesis, 246 150 Linnik's constant, 279-280 Guinness Book oj Records, 1 Linnik's theorem, 281 log log philosophy, 413-414 Lucas numbers, 55 Heuristic, word, 371 Fibonacci numbers and, 70 Hilbert's tenth problem, 188 Lucas pseudoprimes, 126-135 Hilbert-Waring's identity, 301- distribution of, 317-318 302 Lucas sequences, 53-74 Hurwitz, idea of, 4-5 associated to pairs, 55 companion, 55 primality tests based on, 74-83 Ideal numbers, 324 Lucas's test, 49-50 Index of notations, xvii-xxiv Infinite sequences of pairwise relatively prime numbers, Mann and Shanks, property of, 29 generation of, 17-18 Mascheroni's constant, 222 Irregularity index, 328-329 Mersenne numbers, 90-103 Iterated gaps between prime completely factored, 97 numbers, 257-258 composite, 378 primality tests for, 92-93 Mersenne primes, distribution Jacobisymbol,47-48 of, 411-413 lacobsthal function, 280 Mertens function, 230 Metrod's proof, 9 Mirimanoffs congruence, 334 Knodel numbers, 125-126 Mobius function, 182-183 Kulik's table, 233-234 sums involving, 229-233 Kummer's lemma on units, 325 Mobius inversion formula, 230 Kummer's proof, 4 Multifactorials, 27 Kummer's theorem, 32 Multiple perfect numbers, 103 Multiplicity of m, 39

Largest prime factor, 45-46 Law of appearance of p, 64 Notations, index of, xvii-xxiv Law of repetition, 65 NSW primes, 367-369 Legendre's theorem, 30 Number field sieve, 165 538 Subject Index

Odd Goldbach problem, 292- Primality tests 293 with Abelian varieties, 152 Odd Goldbach statement, 291 APR test, 146-147 Order of a modulo n, 35 based on Lucas sequences, Order of a modulo p, 22 74-83 Ore's conjecture, 101-102 classical, based on congru- ences,48-53 cost of, 136-138 Palindromic numbers, 159-160 with elliptic curves, 148-152 Partitio Numerorum, 403-411 general purpose, 139 Pell numbers, 72 large prime numbers and, Pell sequences, 55 135-162 Pentagonal numbers, 308 for Mersenne numbers, 92- Pepin's test, 84-85 93 Perfect numbers, 98-103 running time of, 153 multiple, 103 special purpose, 139 odd,99 Prime factors Perott's proof, 8 largest, 45-46 Perrin pseudoprimes, 135 primitive, 43-45, 67 Polignac's conjecture, 264-265 Prime number(s), 1 Polynomials in arithmetic progression, of arbitrary degree, prime 265-287 numbers as values of, 386- consecutive, 287 400 curious, 159-162 cubic,206 distribution of, 213-322 cyclotomic, 267 expressing in terms of preced- linear, prime numbers as val­ ing prime, 181-182 ues of, 372-386 families of, 323-369 with many initial prime abso­ of form m2 + 1, 206-207 lute values, 197-203 formula for nth, 182-186 with many successive compos­ functions defining, 179-212 ite values, prime numbers gaps between, 250-259 in, 400-403 generation of infinite se- prime-producing, 196-212 quences of pairwise rela­ Poulet number, 106 tively, 17-18 Powerful numbers, 340 gigantic, 159 Primality, 19-20 with given initial and final dig­ Primality certification, 153- its, 355 155 heuristic and probabilistic re­ Primality certification algo­ sults about, 371-425 rithm, 154-155 infinitely many, proofs for, Subjec:t Index 539

see Proofs for infinitely as values of linear polynomi­ many prime numbers als, 372-386 iterated gaps between, 257- as values of polynomials of ar­ 258 bitrary degree, 386-400 large Wieferich, 333-349 fast generation of, 156-157 Prime number set as diophan­ primality tests and, 135-162 tine set, 190 largest gap between, 251 Prime number theorem largest known, 95, 158 citations for work on, 429- largest known of form p# + 430 1, 16 fundamental, 225-227 largest prime of form p# - 1, Prime-producing polynomials, 16 196-212 multifactorial, 27 Prime-producing polynomials NSW, 367-369 races, 203-206 nth, 248-249 Primes, see Prime number(s) number of, 3-18 Primitive polynomials, 196 oddest, 2 Primitive prime factors, 43-45, in polynomials with many suc­ 67 cessive composite values, Primitive root modulo p, 22 400-403 Primordial of p, 12 powers of, dividing factorials, Probabilistic estimates, 411-414 30-33 Probabilistic methods, 371 probable, 140 Probable prime numbers, 140 recognizing, 19-178 Proofs for infinitely many prime records on, 2; see also Re- numbers cords Auric's, 9 regular, see Regular primes Euclid's, 3-4 and second-order linear recur­ Euler's, 6-7 rence sequences, 361-367 Furstenberg's, 10--11 smallest, in arithmetic pro­ idea of Hurwitz, 4-5 gression, 277-284 Kummer's, 4 in special sequences, 288- Metrod's,9 290 Perott's, 8 strings of, in arithmetic pro- Schorn's, 5 gression, 284-287 Stieltjes', 4 tables of, 233-235 Thue's, 7 titanic, 157-159 Washington's, 9-10 topics in, 2 Proth's criterion, 52 twin, 259-265 Pseudoprimes, 103-118 up to 10,000,513-515 distribution of, 311-314 540 Subject Index

Pseudoprimes (Cont.) largest known primes, 95, 158 elliptic, see Elliptic pseudo- largest not Mersenne number, primes 356 Euler, 112-113 largest of form k x 2n + 1, Euler-Lucas, 130-131 356 even, 107 largest prime of form n! + 1, Fibonacci, 127-129 26 Lucas, see Lucas pseudo- largest prime of form p# - 1, primes 16 new kinds of, 134-135 largest prime of form p# + 1, Perrin, 135 16 Somer, 117 Mersenne prime, 95 Somer-Lucas, 131-132 palindromic primes, 159- strong, 113-117 160 strong Lucas, 130-131 for q = 41,200 Pseudoprime tests, 140-146 quadratic polynomial, 204- Public key cryptography, 173- 206 178 regular primes, 326, 327 repunits, 351 for Ruby's polynomials, 203 Quadratic residues, 46-48 Sierpinski number, 357-358 smallest Fermat numbers, 88 Sophie Germain primes, 330- Real quadratic fields, 212 331 Records string of primes in arithmetic Carmichael number, 121 progression, 286 completely factored Mersenne for 0, 254 number, 97 twin primes, 262, 263-264 composite Fermat number, 86 Wieferich primes, 334, 337 composite Mersenne number, Wilson primes, 346, 350 96 zeros, 241 Cunningham chains, 333 References curious primes, 161-162 general, 433-434 Fermat prime, 86 specific, 434-507 generalized Fermat numbers, Regular primes, 323-329 359-360 density of set of, 414-425 for Goldbach's conjecture, Repunits, 350-354 299 r-gonal numbers, 309 largest computed value of Riemann function, 224-225 7r(x),236 Riemann's formula, 224 largest gap between prime Riemann's hypothesis, 139,223, numbers, 251 232 Subject Index 541

Riemann zeta function, 223 Thue's proof, 7 Riesel numbers, 357-358 Titanic primes, 157-159 RSA-system, 173 Totient, 34 Ruby's polynomials, 203 Triangular numbeTs, 308 Tschebycheffs function, 221 Twin-prime constant, 262 Schnirelman constant, 293 Twin primes, 259--264 Schorn's proof, 5 Second-order linear recurrence sequences, prime numbers Uniform distribution, 288-290 and, 361-367 Universal diophantine polyno­ Sequences of binomials, 42-46 mial, 194-195 Sierpinski numbers, 357-358 Sieve of Eratosthenes, 20-21 Solovay and Strassen's method, Valence function, 39 143 Vinogradov's theorem, 292 Somer-Lucas pseudoprimes, von Mangoldt function, 226 131-132 Somer pseudoprimes, 117 Sophie Germain primes, 91, Ward's test, 82-83 329-333 Waring-Goldbach problem, Sophie Germain theorem, 329 310-311 Special purpose factoring algo- Waring's problem, 300-309 rithms, 165-166 Washington's proof, 9-10 Specific references, 434-507 Wieferich primes, 333-349 Square numbers, 308 Wilson primes, 346, 350 Stieltjes' proof, 4 Wilson quotient, 346 Stirling's formula, 218 Wilson's theorem, 25-27, 48 Strong elliptic pseudoprimes, Wolstenholme, property of, 134 29 Strong Lucas pseudoprimes, 130-131 Strongly primitive polynomials, Zero-free regions, 243-245 196 Zeros, 239-243 Strong prime, 157 Zeta function, 216 Strong pseudoprimes, 113-117 Zsigmondy's theorem, 68 Strong pseudoprime tests, 143- 146 Sylvester cyclotomic numbers, 82