Glossary of Symbols a1 ≡ a2(modb) a1 congruent to a2, modulo b; a1 − a2 divisible by b C the field of complex number d(n) the number of (positive) divisors of n; σo(n) d|nddivides n; n is a multiple of d; there is an integer q such that dq = n d | nddoes not divide n e base of natural logarithms; 2.718281828459045... exp{} exponential function 2n Fn Fermat numbers: 2 + 1 Or Fibonacci numbers f (x)=0(g(x)) f (x)/g(x) → 0asx → ∞ f (x)=0(g(x)) there is a constant c such that | f (x)| < cg(x) i square root of −1; i2 = −1 lnx natural logarithm of x (m,n) GCD (greatest common divisor) of m and n; highest common factor of m and n 405 406 Glossary of Symbols [m,n] LCM (least common multiple) of m and n.Also, the block of consecutive integers, m,m + 1,...n p Mp Mersenne numbers: 2 − 1 n! factorial n; 1 × 2 × 3 × ...× n n k n choose k; the binomial coefficient n!/k!(n − k)! p or (p/q) Legendre symbol, also fraction q panpa divides n,butpa+1 does not divide n pn the nth prime, p1 = 2, p2 = 3, p3 = 5,... Q the field of rational numbers rk(n) least number of numbers not exceeding n, which must contain a k-term arithmetic progression x Gauss bracket or floor of x; greatest integer not greater than x x ceiling of x; last integer not less than x xn least positive (or nonnegative) remainder of x modulo n Z the ring of integers Zn the ring of integers, 0, 1, 2,...,n − 1 (modulo n) γ Euler’s constant; 0.577215664901532... π ratio of circumference of circle to diameter; 3.141592653589793... π(x) number of primes not exceeding x, also primitive polynomial Glossary of Symbols 407 π(x;a,b) number of primes not exceeding x and congruent to a, modulo b ∏ product σ(n) sum of divisors of n σk(n) sum of kth powers of divisors of n ∑ sum φ(n) Euler’s totient function; number of positive integers not exceeding n and prime to n ω complex cube root of 1, ω3 = 1, ω = 1, ω2 + ω + 1 = 0 ω(n) number of distinct prime factors of n Ω(n) number of prime factors of n, counting repetitions References Chapter 1 1.1 T. M. Apostol: Introduction to Analytic Number Theory (Springer, Berlin, Heidelberg, New York 1976) 1.2 I. Asimov: Asimov on Numbers (Doubleday, Garden City, NY, 1977) 1.3 A.O.L. Atkin, B.J. Birch (eds.): Computers in Number Theory (Academic, London 1971) 1.4 E. R. Berlekamp, J. H. Conway, R. K. Guy: Winning Ways (Academic, London 1981) 1.5 W. Kaufmann-Buhler:¨ Gauss. A Biographical Study (Springer, Berlin, Heidelberg, New York 1981) 1.6 P.J. Davis: The Lore of Large Numbers (Random House, New York 1961) 1.7 L. E. Dickson: History of the Theory of Numbers, Vols.1–3 (Chelsea, New York 1952) 1.8 U. Dudley: Elementary Number Theory (Freeman, San Francisco 1969) 1.9 C. F. Gauss: Disquisitiones Arithmeticae [English transl. by A. A. Clarke, Yale University Press, New Haven 1966] 1.10 W. Gellert, H. Kustner,¨ M. Hellwich, H. Kastner¨ (eds.) The VNR Concise Encyclopedia of Mathematics (Van Nostrand Reinhold, New York 1977) 1.11 R. K. Guy: Unsolved Problems in Intuitive Mathematics, Vol.I, Number Theory (Sprin- ger, Berlin, Heidelberg, New York 1981) 1.12 H. Halberstam, C. Hooley (eds.): Progress in Analytic Number Theory, Vol.I (Academic, London 1981) 1.13 G. H. Hardy: A Mathematician’s Apology (Cambridge University Press, Cambridge 1967) 1.14 G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, 4th ed. (Clarendon, Oxford 1960) 1.15 L. H. Hua: Introduction to Number Theory (Springer, Berlin, Heidelberg, New York 1982) 1.16 K.-H. Indlekofer: Zahlentheorie, Uni-Taschenbucher¨ 688 (Birkhauser,¨ Basel 1978) 1.17 K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory (Springer, New York 1990) 1.18 H. Minkowski: Diophantische Approximationen (Teubner, Leipzig 1907; reprinted by Physica, Wurzburg¨ 1961) 1.19 T. Nagell: Introduction to Number Theory (Wiley, New York 1951) 1.20 C. S. Ogilvy: Tomorrow’s Math (Oxford University Press, Oxford 1962) 1.21 O. Ore: Number Theory and Its History (McGraw-Hill, New York 1948) 1.22 H. Rademacher: Lectures on Elementary Number Theory (Blaisdell, New York 1964) 1.23 H. Rademacher, O. Toeplitz: The Equipment of Mathematics (Princeton University Press, Princeton 1957) 409 410 References 1.24 A. Scholz, B. Schoenberg: Einfuhrung¨ in die Zahlentheorie, Sammlung Goschen¨ 5131 (Walter de Gruyter, Berlin 1973) 1.25 C. E. Shannon: Communication theory of secrecy systems. Bell Syst. Tech. J. 28, 656–715 (1949) 1.26 W. Sierpinski:´ 250 Problems in Elementary Number Theory (American Elsevier, New York 1970) 1.27 J. V. Uspensky, M. A. Heaslet: Elementary Number Theory (McGraw-Hill, New York 1939) 1.28 D. J. Winter: The Structure of Fields, Graduate Texts in Mathematics, Vol.16 (Springer, Berlin, Heidelberg, New York 1974) Chapter 2 2.1 L. H. Hua: Introduction to Number Theory (Springer, Berlin, Heidelberg, New York 1982) 2.2 R. Plomp, G. F. Smoorenburg: Frequency Analysis and Periodicity Detection in Hearing (A. W. Sijthoff, Leiden 1970) Chapter 3 3.1 C. Pomerance: The search for prime numbers. Sci. Am. 247, No. 6, 136–147 (1982) 3.2 W. H. Mills: A prime representing function. Bull. Am. Math. Soc. 53, 604 (1947) 3.3 T. Nagell: Introduction to Number Theory (Wiley, New York 1951) 3.4 D. Slowinski: Searching for the 27th Mersenne prime. J. Recreational Math. 11, 258–261 (1978–79) 3.5 D. B. Gillies: Three new Mersenne primes and a statistical theory. Math. Comp. 18, 93–97 (1963) 3.6 G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, 5th ed., Sect. 2.5 (Clarendon, Oxford 1984) 3.7 W. Kaufmann-Buhler:¨ Gauss. A Biographical Study (Springer, Berlin, Heidelberg, New York 1981) 3.8 C. Chant, J. Fauvel (eds.): Science and Belief (Longman, Essex 1981) Chapter 4 4.1 P. Erdos,¨ M. Kac: The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62, 738–742 (1945) 4.2 P.D.T.A.Elliot: Probabilistic Number Theory, Vols.1–2 (Springer, Berlin, Heidelberg, New York 1980) 4.3 D. Zagier: “Die ersten 50 Millionen Primzahlen” in Lebendige Zahlen, ed. by F. Hirze- bruch (Birkhauser,¨ Basel 1981) 4.4 G. Kolata: Does Godel’s¨ theorem matter to mathematics? Science 218, 779–780 (1982) 4.5 P. Erdos:¨ On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A. 35, 374–384 (1949) 4.6 H. M. Edwards: Riemann’s Zeta Function (Academic Press, New York 1974) 4.7 Z. Furedi,¨ J. Komlos: The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981) 4.8 M. R. Schroeder: A simple function and its Fourier transform. Math. Intelligencer 4, 158–161 (1982) 4.9 U. Dudley: Elementary Number Theory (Freeman, San Francisco 1969) 4.10 G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, 5th ed. Sect. 22.3 (Clarendon, Oxford 1984) 4.11 M. R. Schroeder: Speech Communication 1, 9 (1982) 4.12 I. M. Vinogradov: An Introduction to the Theory of Numbers (Pergamon, New York 1955) References 411 Chapter 5 5.1 C. D. Olds: Continued Fractions (Random House, New York 1963) 5.2 H. S. Wall: Analytic Theory of Continued Fractions (Van Nostrand, Princeton 1948) 5.3 A. N. Khovanskii: The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory (Noordhoff, Groningen 1963) 5.4 A. Y. Khinchin: Continued Fractions (University of Chicago Press, Chicago 1964) 5.5 F. D. M. Haddani: Phys. Rev. Lett. 51, 605–607 (1983) 5.6 K. Ikeda, M. Mitsumo: Phys. Rev. Lett. 53 1340–1343 (1984) 5.7 C. J. Bouwkamp, A. J. Duijvestijn, P.Medema: Tables relating to simple squared rectan- gles (Dept. of Mathematics and Mechanics, Technische Hogeschool, Eindhoven 1960) 5.8 V.E. Hoggatt: Fibonacci and Lucas Numbers (Houghton Mifflin, Boston 1969) 5.9 P.H. Richter, R. Schranner: Leaf arrangment. Naturwissenschaften 65, 319–327 (1978) 5.10 M. Eigen: “Goethe und das Gestaltproblem in der modernen Biologie,” in H. Rossner¨ (ed.): Ruckblick¨ in die Zukunft (Severin und Siedler, Berlin 1981) 5.11 O. Ore: Number Theory and Its History (McGraw-Hill, New York 1948) 5.12 A. Koenig (personal communication) 5.13 W. Gellert, H. Kustner,M.Hellwich,H.K¨ astner¨ (eds.): The VNR Concise Encyclopedia of Mathematics (Van Nostrand Reinhold, New York 1977) 5.14 L. K. Hua, Y. Wang: Applications of Number Theory to Numerical Analysis IX (Springer, Berlin, Heidelberg, New York 1981) 5.15 J. C. Lagarias, A. M. Odlyzko: Solving “low-density” subset sum problems. J. Association of Computing Machinery 32, 229–246 (1985) 5.16 R. L. Graham (personal communication) 5.17 R. K. Guy: Unsolved Problems in Intuitive Mathematics, Vol.I, Number Theory (Sprin- ger, Berlin, Heidelberg, New York 1981) 5.18 M. Gardner: Mathematical games. Sci. Am. 239, No. 4, 22–26 (1978) 5.19 R. L. Graham: A theorem on partitions. J. Austral. Math. 4, 435–441 (1963) 5.20 E. Landau: Elementary Number Theory (Chelsea, New York 1958) 5.21 E. H. Neville: The Farey Series of Order 1025 (Cambridge University Press, Cambridge 1950) 5.22 C. M. Rader: Recovery of undersampled periodic waveforms. IEEE Trans.