MATHEMATICSOF COMPUTATION VOLUME64, NUMBER 212 OCTOBER 1995, PAGES 1733-1741
NEW CULLEN PRIMES
WILFRID KELLER
Abstract. Numbers of the forms C„=«-2" + l and W„ = n-2" -\ are both called Cullen numbers. New primes C„ are presented for n = 4713 , 5795 , 6611 , 18496 . For Wn , several new primes are listed, the largest one having n = 18885 . Furthermore, all efforts made to factorize numbers Cn and W„ are described, and the result, the complete factorization for all n < 300, is given in a Supplement.
1. Introduction In 1905, the Reverend J. Cullen [6] called attention to the numbers Cn = n • 2" + 1, the particular case of k • 2" + 1 where k - n . He observed that C„ was composite for all « in the interval 1 < n < 100, with the only possible exception of « = 53 . Shortly afterwards, Cunningham [7] gave the prime factor 5591 for C53 and restated Cullen's assertion for 1 < « < 200, now leaving « = 141 as the only uncertain case in the considered range. Cunningham pointed out that primes of the form C„ = « • 2" + 1 seemed to be remarkably rare. Haifa century later, in 1957, Robinson [22] showed that CX4Xwas in fact a prime. Moreover, he established that this was the only prime C„ with 1 < « < 1000. In our numerical investigation we were able to determine four additional primes Cn . It can now be asserted that Cn is prime for « = 1, 141, 4713, 5795, 6611, 18496, and for no other « < 30000. The analogous numbers W„ = « • 2" - 1 have also been investigated. As to their primality, Riesel [21] found in 1968 that Wn is prime for « = 2, 3, 6, 30, 75 , 81, but for no other « < 110. Considerably extending this search, we could add to Riesel's list the primes Wn with « = 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885. All other values of « < 20000 yielded composite numbers. A comprehensive study of divisibility properties of numbers C„ and W„ was presented by Cunningham and Woodall (abbreviated C&W in what fol- lows) in 1917. On account of their classical paper [8], the notations C„ (for "Cunningham numbers") and W„ (for "Woodall numbers") have been used, while the term Cullen numbers is extended to both these varieties (cf. [13, 24] ). C&W also initiated the project of actually factoring the considered numbers. This work has been extended over the past decades, inasmuch as increasingly Received by the editor December 3, 1992 and, in revised form, March 28, 1994. 1991 Mathematics Subject Classification. Primary 11A51; Secondary 11B99, 11-04. Key words and phrases. Cullen numbers, factoring, large primes.
©1995 American Mathematical Society
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powerful factorization methods were developed. In a Supplement to this paper the complete factorization of numbers Cn and W„ is given for all « < 300.
2. Factor tables C&W's report includes the factorization of Cn and Wn for all « < 32, except for the single number W30, whose primality was not recognized. In the interval 32 < « < 66, the character of Wn remained doubtful only for « = 39, 42, 51. The factor table for the original Cullen numbers C„ has subsequently been completed at least up to n < 43 by Beeger [1], to n < 61 by E. Karst, D. H. Lehmer and J. S. Madachy (as reported in [13]), and to « < 101 by Steiner [24]. Here, three errata have to be observed: one digit is missing in each of the factorizations given for « = 91, 97, 101 . Also from [13] we learn that the factorizations of Wn were completed for all « < 49 by Karst and Madachy, and that Madachy also factored Wn for « = 52, 55, 56, 57, 59, 60, and 76. The decomposition of W50attributed to Lehmer gives the cofactor 1197765858343217= 30503527 - 39266471 as a prime, and the numbers Wx23 and W249 are falsely reported as composites. The factors of W5Xwere later given in [24]. Finally, [13] lists small factors of numbers Cn and W„ up to « = 300. In the Supplement, the complete factorization of both Cn and Wn is ex- tended to just that limit. The tables were compiled and carefully checked by the author of this paper. Factors for « < 210 considered difficult to obtain at the time were provided by G. Löh and W. Niebuhr. Thus, Löh completed Cn for 102 < « < 171 and Wn to « < 122 in 1984, using the methods of [2, 18, 20], and Niebuhr completed Cn and Wn for « < 210 in 1986 with his implementation of the multiple polynomial quadratic sieve (MPQS) based on the description given in [23]. In 1988, H. Suyama supplied the smallest factor of W21$,and the 16-digit prime factor of W2XX,enabling us to prove primality for the 51-digit cofactor of this number. Suyama used his implementation of the elliptic curve method (ECM), cf. [17]. With only the exception of W2XX,the complete range of 210 < « < 260 was done by the author in 1991, using the excellent factorization and prime-testing routines distributed with version 8.15 of Y. Kida's UBASIC, as released in December 1990. For a description of UBASIC, see [19]. The programs ECMX, MPQSX, and MPQSHD, written by Kida, were run on an IBM PS/2-70 386 PC equipped with a mathematical coprocessor. The most demanding application, the decomposition of the 78-digit number C25i into its two prime factors with MPQSHD, required 264 hours. The completion of the final segment 260 < n < 300 seemed too hard for UBASIC and was therefore not attempted, leaving unfinished 16 numbers C„ and 12 numbers W„. They were finally accomplished with the assistance of Niebuhr and R. P. Brent. Niebuhr factorized 21 of the remaining composites, using his own programs for ECM and for MPQS. The latter was run on an IBM ES/9021-440 mainframe. The largest numbers thus treated were the 77-digit cofactors of C26i, W2(>4, W2(sl, and W2S9,which required about 64 hours of CPU time on the average, and the 79-digit cofactor of C297, which took 81 hours. The other seven composites had 80, 81, 83, or 86 digits. All these were
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attacked and finished with Brent's program MVFAC [4], a vectorized imple- mentation of ECM. For details, see [3] and [17]. The program was run by Brent himself on a Fujitsu VP 2200/10 vector processor at the Computer Sciences Laboratory of the Australian National University, and, more intensively, on an SNI S100/10 (similar to a VP 2100/10) recently installed at the Computation Center of the University of Hamburg. Brent obtained the factorization of W211 (83-digit cofactor), and the author finished the remaining six numbers. The most notable factor determined with MVFAC was a 39-digit prime factor of C296 found after trying only 750 curves. It is also the largest "penultimate" factor contained in Tables I or II of the Supplement. The factorizations for n < 300 were completed on August 16, 1992. Table I of the Supplement includes the factorization of C]2g = 2135+ 1 obtained by Le Lasseur and published by Lucas [16] in 1878 (one digit in the largest factor misprinted), as well as Robinson's prime CX4X.Table II includes the factorization of WM = 270 - 1 already contained in [16], that of WX2S= 2135- 1 established by P. Poulet in 1946 (see [14] ), and the previously known primes. It also includes the factorizations of WXooand W25(,,which, according to [13], had been completed by Karst and K. R. Isemonger, respectively. It should be noted that for « = 100, 144, 196, 256 the factorization of W„ was facilitated by an algebraic decomposition, as was the case historically for Ci28 and Wx2%. C&W listed all 44 values of « < 1000 for which no factor of C„ was known and they showed that in every case the smallest factor p had to be > 1000. Precisely these C„ were tested for primality by Robinson to discover that only G 4i was prime in that range. Apart from this prime, six more of the C„ in question, corresponding to n = 233 , 245 , 251, 252, 285, 293, are in Table I and thus completely factored. In seven cases with « > 300 the factorization of C„ could also be established: C3i8- 10939-100429275849522701-p78, C436= 1081501579•43008589651•3717967975567-pl02, C579= 4988803100279081295749323-pl53, C634= 2459• 1085629591■2756181749•7148901709-pl62, C753= 164834525239-p219, C921= 2428711 - 2719027•4410839• p261, C933= 197724478669-p273.
Here pA denotes an A-digit prime. The seven cofactors, like others to be mentioned below, were proved prime by using the procedure APRT-CL (Cohen- Lenstra version of Adleman-Pomerance-Rumely Test), programmed by K. Aki- yama, which is included in Kida's UBASIC. The factorization of C579 was obtained through MVFAC. For 27 of the remaining 30 Cullen numbers Cn , we give in Table 1 a prime factor. If by trial division a factor p was found below 107, the smallest factor of C„ is tabulated. If, however, a factor p > 107 is given, it was found by one of Pollard's methods or (for « = 581, 648, 941 ) by Brent's MVFAC and might not be the least. Thus, only three Cullen numbers Cn with « < 1000 have no known factor, which are for « = 435, 453, 915 .
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n p n p
333 2423 711 367957 402 1117 713 3079 412 1091 778 33667759 473 365969 816 6451 516 20641 849 7103923 532 18313 869 69508729 533 95257 870 39869 580 2843 899 1633716607 581 2660379251641 900 18176209 587 60744852593 916 247451 588 203793838081 941 7717335184972583304615708011 609 70038149 942 2598767 648 240383530451966593 953 381287 693 134409623339
A prime factor p < 107 does exist for all but 41 of the composite numbers W„ with « < 1000. Six of these are completely factored since they have « < 300. The following six with « > 300 were factored by finding one factor (in the case of Hg85, MVFAC determined a composite factor): W369= 7728415141 -pl04, 1^463= 141959514756636037-pl25, W612= 962794776758617477606791593-pl79, Wn9 = 738497627270005859999837• p217, Wm = 8353578155864671•84769280380290403-p237, W90S= 3057961301 - p267. For 23 of the remaining 29, a factor p > 107 is presented in Table 2. The factor given for « = 366 and the smaller factor of W4¿3 were kindly supplied
Table 2. Prime factors p of Cullen numbers W„ = « • 2" - 1
332 12165323 675 1608969227141 350 5169607633 722 64952161193 366 62497690394803 723 128915821 386 119570203 765 130234223449138177 423 3537292571 795 80958347 522 40818521 824 1757762099 541 3241623612714017 866 10885241 564 26768197513 906 16556069 570 48623921 931 15127751 621 16127089673 932 114082154860229 663 60166683064673819 943 405108238890652513 669 73399869877
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NEW CULLEN PRIMES 1737 by H. Suyama, who found them in 1988. Those for « = 541, 663, 765, 932, 943 are due to MVFAC, like the above factors of W672and Wn9 . The numbers Wn without a known factor now occur for « = 349, 375, 668, 715, 951, 963. There is presently a project underway attempting complete factorizations for the whole range 300 < n < 1000. As of the end of February 1994, 147 numbers Cn and 142 numbers Wn had virtually been finished. A machine readable list with all known factors is available from the author, who would welcome any new factors that readers may wish to contribute.
3. Divisibility properties In pursuance of the original observations made by Cullen and by Cunning- ham, a glance at the factor table for Cn suggests that a prime p divides both the numbers Cp-X and Cp-2. A more general statement is in fact true. For a given odd prime p let nk = (2k -k)(p - 1) - k , k > 0. This includes «o = p - 1 and «i = p-2. The statement is that p divides C„k for all k > 0, and it is verified easily. Since nk = -2k (mod/?), we have nk • 2"" = -2k+n« = -2^-k^-X) (mod p), which by Fermat's theorem is congruent to -1 modulo p . Therefore C„k = 0 (mod p). Obviously, nk cannot be a multiple of p. To describe the totality of numbers C„ that are divisible by a given prime p, two remarks are in order. First, the numbers Cn mod p are periodic with period php, where hp = expp(2) is the smallest positive integer « such that 2h = 1 (mod p). As a consequence, if p divides C„ for some «, then p also divides Cn+pn . Secondly, from the definition of nk it may be deduced that there are exactly hp numbers nk which are incongruent modulo php , and no other « with 0 < « < php gives a C„ with a prime factor p (cf. [8]). So, the totality of all « such that Cn is divisible by p is obtained as follows. Determine n'k —nk mod php for k = 0, 1, ... , hp - 1, 0 < n'k < php . Then, for every k , include « = n'k + rphp for all r > 0. In particular, the prime p = 3 divides all C„ with « = 1,2 (mod 6). As a further example, for p = 11, hp = 10, we get n'k = 10, 9, 18, 47, 6, 45, 24, 103, 52, 71, and all integers n congruent modulo 110 to any one of these. A similar description can be given for the set of subscripts « for which W„ is divisible by p . Let nk = - (2k + k)(p - 1) - k (these are negative integers) for k > 0, determine n'k = nk modphp for k = 0, 1, ... , hp - 1, 0 < n'k < php , and, for every k, include n = n'k + rphp for all r > 0. In this case, p = 3 divides all W„ with « = 4,5 (mod 6), and p = 11, hp = 10 give n'k = 100, 79, 48, 107, 16, 65, 64, 73, 102, 61, and all « congruent modulo 110 to any one of these. Note that for « = 65, 64 we have two consecutive numbers Wn , and ac- tually an infinity of such pairs, which are both divisible by the same prime p = 11. For numbers C„ this situation occurs for every prime p , as we have seen. For numbers Wn, however, this is restricted to the case that hp is an even number. The minimality of hp then implies that 2a*1/2= -1 (mod p) and thus also 2ph"/2= -1 (mod p). It is now easily seen that W„ = 0 (mod p) for n = php¡2 + p - 1 and for « = php¡2 + p - 2. Other divisibility rules are also contained in the given general description.
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For instance, p divides C^p+X)/2and W^3p_Xy2,or it divides C(3P_i)/2 and ^(p+i)/2 >according as the Jacobi symbol (2/p) is -1 or +1. Again, this can be verified immediately. For more details, see [7] and [8]. In spite of the apparent similarity in the description of small factors for both sequences {C„} and {Wn}, considerably more composites are obtained for the original Cullen numbers Cn . This is primarily due to the fact that every prime p > 3 immediately gives four consecutive composite numbers C„ . In addition to Cp-.x and Cp-2, always two neighboring numbers of the sequence are divisible by 3. If p is of the form p = 6k + 1, then 3 divides Cp and Cp+X, and in the case of p = 6k - 1, the prime 3 divides Cp-3 and Cp-4. In the special situation of a twin prime pair p, q = 6k ± 1, together a string of eight consecutive composites C„ occurs. To give an indication of how the notable difference in the number of primes of the forms Cn and Wn comes about, let us first consider all values of « < 110 such that C„ has no prime factor p < « + 2. Here we get « = 33, 53 , 75 , where « = 75 can be eliminated through the prime factor p = 2« - 1 = 149. In the case of W„ , however, 32 values of « remain in the first step, of which eight are eliminated by a factor p = 2« - 1 . The remainder of 24 includes, of course, the six values of « < 110 giving rise to the primes W„ determined by Riesel. 4. Cullen primes In our search for new primes, we tested C„ for « < 20000 in 1984 and continued (without success) to « < 30000 in 1987. Similarly, Wn was tested up to « < 15000 in 1984, and to « < 20000 in 1987. In this case two more primes were found for « = 15822 and « = 18885. All other new Cullen primes were thus discovered in 1984. The methods used for proving primality are found in [5]. For 1000 < « < 20000, a total of 632 numbers C„ and 1203 numbers W„ had actually to be tested after the sieving procedure. In addition, 321 numbers C„ with 20000 < n < 30000 remained without known factor. We also considered the possibility of « • 2" ± 1 forming a pair of twin primes. Such "Cullen twins" may indeed exist for some large «. But we showed (by testing individual numbers Cn ) that this « must exceed « = 41528 . The prime C18496= 172-218502+1 = (17-29251)2+1 might be of some interest in relation to Conjecture E of Hardy and Littlewood [10], which states that an infinity of primes A2 + 1 should exist (actually, the conjecture was given in a quantitative asymptotic form). At the time of its discovery, our prime seemed to be the largest one known of that expression. But now we know five larger ones, discovered by H. Dubner (unpublished), which are of the form b2m+ 1, with b even. The largest of these primes is 2009442 + 1, which has 10861 digits and was found in September, 1992. Incidentally, Dubner [9] has also considered a generalization of Cullen numbers given by « • b" + 1 for b > 2 . Regarding the primes W„ listed for « > 110, it appeared that four of them had already been known for some time. As a matter of fact, Williams and Zarnke [25] had found Wxxsto be prime, Jönsson [12] had found 181-2363 — 1 = W3(s2,Riesel [21] had given 3 • 2391- 1 = W3%4, and W5X2turned out to be the Mersenne prime M¡2X= 2521- 1 discovered by Robinson (see [15]). A first result concerning the distribution at large of Cullen primes Cn was obtained by Hooley [11]. By applying sieve methods, he showed that the natural
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density of positive integers « < x for which C„ is a prime is of the order o(x) for x —*oo. In that sense, almost all Cullen numbers are composite. Suyama (communicated personally) has reworked Hooleys proof to show that it is applicable to any sequence of numbers « • 2n+a + b, where a, b are arbitrarily fixed integers. In particular, Hooley's result also follows for numbers Wn. In the context of a search for prime factors of Fermât numbers our particular interest in Cullen numbers C„ emanated from the remark of C & W, supported by an explicit argument, that primes of the form C„ seemed likely as possible divisors of Fermât numbers. However, the four new primes C„ proved not to divide any of these. Instead, there is some evidence for the likelihood that a prime W„ divides a Mersenne number Mp . We noticed that W2 = M3 and Wix2 = Ms2x, and that W3 divides Mxx, W6 divides MX9X,and WX23divides Mp for p = 123 • 2122- 1, a 39-digit prime. In all three cases where W„ properly divides Mp , we have p = (Wn-l)/2 = «-2"~'-l, that is, p and the divisor Wn = 2p+l are both primes. To establish in general whether W„ divides any Mp or not, the complete factorization of (W„- l)/2, if available, can be used. In fact, a divisor Wn of Mp must also be of the form 2kp + 1 . Equivalently, ( Wn- l)/2 = kp , hence the particular subscript p should divide (W„ - l)/2. So, for all different prime factors p of this number, one would have to check if Mp mod W„ = 0. But in practice a much easier test suffices, as Suyama has indicated in a personal communication. To see whether a prime q divides some Mersenne number or not, only an "evenly factored portion" of q - 1 and its cofactor need to be used. Let q be a prime and q - 1 = 2a° • p"[ • p%2.p?r • C, where p¡ are odd primes and C is composite. If 2P>^ 1 (mod q) for 1 < j < r and 2C ^ 1 (mod q), then q cannot be a factor of any Mersenne number. In the alternative case that 2C = 1 (mod q), q may be a factor of Mp , where p is some prime factor of C. Indeed, if for an unknown factor p of C we had q | Mp , that is 2P = 1 (mod q), then necessarily 2C = 1 (mod q). For all primes q = W„ with 30 < « < 822 (except for the otherwise settled « = 123 ) this test was conclusive by using only one odd factor px of Wn- 1. In most cases, a very small px was readily found, while for « = 249 and n = 384 the factors px = 708211533214392631resp. px = 12694590781 determined by Suyama had to be used. It has also been checked that the Cullen number 1^5312does not divide any Mersenne number, by calculating 2C mod ^5312 for C — (^5312-1)/(2-3-52). But here we had the interesting case that, nevertheless, 2C> = 1 (mod ^5312) for C = (W53l2 - l)/(2 • 3). For ^7755, the test could not be carried out, because no factor of the composite number (^7755 - l)/2 was found. Recently, Dubner has kindly tested the numbers W„ for « = 9531 , 12379, 15822, 18885, without detecting a divisibility. In the case of n = 15822 he obtained 2C = 1 (mod Wl5S22),where C = (Wi5m - l)/(2 • 7) and no further factor of this number was known. Of course, W„ may also be identical to a Mersenne number without being a prime. Generally, Wn is a Mersenne number whenever « = 2m and m + 2m is prime. This is the case for m = 1 : W2= M3 - 1 ; m - 3 : W%= M\ \ = 23-89 ; m = 5: W32= M31 = 223-616318177; m = 9: H^512= M52, ; and also for m= 15, 39, 75, 81, 89, 317, 701, 735 and no other m < 1310.
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5. Biographical note When we searched the literature for references to Cullen numbers, we were intrigued by the fact that almost nothing could be found about Cullen as a person. Even mathematical or encyclopaedic dictionaries usually giving such information failed to disclose Cullen's complete first name or to mention the year of his birth or his death. Only the abbreviation "S. J." placed after his name in a footnote to [8] suggested that he was a Jesuit priest. That hint finally enabled us to locate the most appropriate and authoritative source for this matter, namely, the Department of Historiography and Archives of the English Province of the Society of Jesus. The archivist of that institution, Father T. G. Holt, was kind enough to provide the following biographical data, which we are pleased to present to the reader in unabridged form.
Father James Cullen, S. J., was born on April 19th 1867 at Drogheda in Ireland. At first he was educated privately, then by the Christian Brothers. Next he went to Trinity College, Dublin, to study pure and applied Mathematics. Afterwards he was in business for a while, but after a short period at an apostolic school in Ireland to learn Latin he entered the noviceship of the English Jesuits at Manresa House, Roehampton, London, as he had decided to become a priest. From 1892 till 1895 he was at Manresa in the noviceship and then for a year's study. From 1895 till 1897 he studied philosophy at the Jesuit house for philosophical studies at St Mary's Hall, Stonyhurst in Lancashire. From 1897 till 1901 he studied theology at the theologate of the English Jesuits at St Beuno's College in North Wales. He was ordained priest at St Beuno's College on July 31st 1901. In 1902 and 1903 he taught Mathematics to young Jesuits who had finished their noviceship at Manresa House. In 1905 he taught Mathematics at the Jesuit boarding school Mount St Mary's College in Derbyshire. It is said that he found difficulty in teaching. In 1906 he was sent to Stonyhurst College, also a boarding school, as accountant and later manager of the College farm and estate. Meanwhile he kept in touch with leading mathematicians and contributed articles to Nature, the Mathematical Gazette and the Messenger of Mathematics. In 1921 he left Stonyhurst and was appointed to supervise accounts of other English Jesuit houses. He died on December 7th 1933.
Acknowledgments The author thanks Hiromi Suyama for his remarks and suggestions indicated in the text, as well as for the provision of some prime factors, and to Günter Löh for the factorizations mentioned in §2. Thanks are also due to Wolfgang Niebuhr for his willingness to produce difficult factorizations whenever asked to do so. He has been appropriately included as a coauthor of the Supplement in acknowledgment of the substantial contributions he has made to it over the years. Special thanks are extended to Richard Brent for making available his program MVFAC and thus allowing us to obtain the last factorizations needed to complete the Supplement.
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Finally, the author is greatly indebted to Father Bruno Brinkman of Hey- ihrop College, University of London, for helping to establish contact with Father Geoffrey Holt, who compiled and wrote the enlightening biographical note of §5-
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1. N. G. W. H. Beeger, Cullen numbers, MTAC 8 (1954), 188. 2. R. P. Brent, An improved Monte Carlo factorization algorithm, BIT 20 (1980), 176-184. 3. _, Some integer factorization algorithms using elliptic curves, Austral. Comput. Sei. Comm. 8(1986), 149-163. 4. _, MVFAC: A vectorized Fortran implementation of the elliptic curve method, Comput. Sei. Lab., Austral. Nat. Univ., 1991. 5. J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factoriza- tions of b"±l, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, 2nd ed., Contemp. Math., vol. 22, Amer. Math. Soc, Providence, RI, 1988. 6. J. Cullen, Question 15897,Educ. Times, Dec. 1905, 534. 7. A. Cunningham, Solution of Question 15897, Math. Quest. Educ. Times 10 (1906), 44-47. 8. A. Cunningham and H. J. Woodall, Factorisation of Q = (2q^q) and (q.2q^\), Messenger Math. 47(1917), 1-38. 9. H. Dubner, Generalized Cullen numbers, J. Recreational Math. 21 (1989), 190-194. 10. G. H. Hardy and J. E. Littlewood, Some problems of'Partitio numerorum'; III: On the expression of a number as sum of primes, Acta Math. 44 (1923), 1-70. U.C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Univ. Press, Cambridge, 1976. 12. I. Jönsson, On certain primes of Mersenne-type, BIT 12 (1972), 117-118. 13. E. Karst, Prime factors of Cullen numbers n*2" ± 1 , Number Theory Tables, compiled by A. Brousseau, Fibonacci Assoc, San Jose, Calif., 1973, pp. 153-163. 14. D. H. Lehmer, On the factors of 2" ± 1 , Bull. Amer. Math. Soc. 53 (1947), 164-167. 15. _, Recent discoveries of large primes, MTAC 6 (1952), 61. 16. E. Lucas, Sur la série récurrente de Fermât, Bull. Bibl. Storia Se. Mat. Fis. 11 (1878), 783-798. 17. P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp. 48 (1987), 243-264. 18. M. A. Morrison and J. Brillhart, A method of factoring and the factorization of F-¡, Math. Comp. 29(1975), 183-205. 19. W. D. Neumann, UBASIC:a Public-Domain BASICfor Mathematics, Notices Amer. Math. Soc. 36 (1989), 557-559; UBASICUpdate, ibid. 38 (1991), 196-197. 20. J. M. Pollard, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc. 76(1974), 521-528. 21. H. Riesel, Lucasian criteria for the primality of N = h • 2" - 1 , Math. Comp. 23 (1969), 869-875. 22. R. M. Robinson, A report on primes of the form k-2" + l and on factors of Fermât numbers, Proc. Amer. Math. Soc. 9 (1958), 673-681. 23. R. D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), 329-339. 24. R. P. Steiner, On Cullen numbers,BIT 19 (1979), 276-277. 25. H. C. Williams and C. R. Zarnke, A report on prime numbers ofthe forms M = (6a+l)22m~' - 1 and M' = (6a - l)22m - 1 , Math. Comp. 22 (1968), 420-422.
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Supplement to NEW CULLEN PRIMES
WILFRID KELLER AND WOLFGANG NIEBUHR
Table I Factorization of Cullen numbers Cn = n • 2" + 1
C„ = n-2" + l 1 3 2 32 3 52 4 5 13 5 7-23 6 5-7-11 7 3•13 23 8 3 683 9 11 • 419 10 72 -11 -19 11 13 ■1733 12 13-19• 199 13 32•11833 14 3-157-487 15 17-29 997 16 17-61681 17 52 • 19 • 4691 18 11 19 107-211 19 3 • 17 • 37 5279 20 33 • 103 ■7541 21 23 • 1914791 22 13 - 23 43 ■7177 23 5 • 151 • 421 ■607 24 5 • 11 • 1399 5233 25 3 ■17 2459 • 6689 26 3•5 • 72 •2373919 27 7 • 29 ■53 ■336823 28 292 • 8937209 29 31-83 6051013 30 172 31 • 59 • 149 ■409 31 32 • 7 61 • 1483 11681 32 3 • 1777• 25781083 33 47 •6031230671 34 19 • 23 ■67 ■1061 • 18803 35 37 •32502455213 36 372•1933-934861 37 3-5 339016085231 38 32 • 20879• 55586743 39 41 • 3433 152326961 40 41 131611-8150491
© 1995 American Mathematical Society
S39
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«o ses I Cî -H « OO Cî
■ too 5§! Ss ! S £2 ! 0O 2 C*J /N5 *—' ^^ 1TJ ^« l— is22-^ w r- irj n t» i'i « r- ■>-■gil° ?s î« W «5 •' p I t- o> £2 M Q P I ; K - « g h n ^ S i r- oo Jr i ce e-5 Î2 r» S 2 « —• o> c?1 Ci ' oo (N ^ S 3^ cn n. r- ' h- 00 Í b-JN Ü CCI ~ S «2 i oo 3 agi: « S 22t " -V A o tO oi * 0 « s ^ S OO h- Tj. CN TT ■ Si' = îi 2 S s Sg , C* Tî- *"* ZÍ S *- S p- - t- r- ' ils §25 2»'g OO' 00 S g SgNoort ) o <à ' S3- 5 fi C s K : 03 s s m ct. -H * 5 S ~* es . es ■*«■ SsE* MOM10 8* r- oo ) S Tí ' * *° ? M ??s: S oo *-. ¡ iS ' w : O »~l ic lis00? iii! liai ï t r; oo > os 3Í r- ¡N ^H O Tp £r S 8 P ! w c< o OO W i :22ï;« -H 00 M i TT i-i Oi tf (û - I OJ 00 ¡Pi 1« ? S Sí ° . o> i *o — "! £■ «*? >• t5 oo« Íi Si ■SS2«3 S-IIsi •Vt- —co ! îs^s- «g - e*s ; S § N _. g § p, 3. Z V g m (g o» <Ç- ! CM — ä s ■« s s ■ PO f- I ■ oo .— X O 3) 0C 0j Cî Q 1 h©« ■-.Mi i r- -h a» ^h •"■»oj cft i eft »> t- cn es . .-■ m o o CO c • ess Oo ■ c* n cî -»r ct> IN - • CC ' ir-r-nnr- m i « mm «' iflN^nONirîWiO in m m if) ^-NMPÎ^iflr-NnwiON v iß w n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SUPPLEMENT S41 Ce (O Ce Ci —i . 00 CO T n w t- - h « n ri m iß < iCNco'Tj'Kî^or-oocj© CSCSCSCÑCÑCÑCÑCÑCÑCÑSCSCNCNCNCNCNCSCNCNCS 1 os i« • ■ «(7 1 imí) ißeoi i H H « P) ce ce •-" - iCNce^lrîïOt^OOOSO' i M m v m License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use S42 SUPPLEMENT giß ^ a. ^ OS_ -*« t= £ S « 3 °> -h te ce iß os es -H CS ¡ ¡ 1¡!¡ t© oo o OS 1 K •» tr os , tft -H os es i t*- io es oo I I ISSi ■V & iß I ° t1- oo t— h- r-H es çs, Iß —1 —« iß , t- es ^, . r- es - , CN OO r- os ce to , «-• es iß es hhcû ; co t ce m to , " wscooio^Nrt^»n^^«0)0'-'csrt'riß'Or-toc>Or-c,irt'fl,iß SCO co 00 3 ce ce a o. S ^ M O iß ; § ^ & ? o. ' Iß t- I aS a. 00 . co o o : |S2?e es ce es o> ce o ce "V 1 2 5 o os ce co 00 t- •— TT OO : 3£S S •- « s os Tf ce « « 1 wS2 00«> esr- ' « r- — es ce iß o — es r- o co o es • ^t 10 ■ 1 00 es ■* t: «çî«i ' —. ce - § S S 1 ■«* t»- ce 90 - co r- L iß OS o E: to es -v ce Tf CO 1 00 O OS 1 1 CO CO o es -fl- 1 t- T iß (O s es ce8 ce' 3!T ■ eo 90 00 O - i OS iß Iß — ' co iß ce S 1 -h ce os os es a 10 -'S« : a"S§; co »ß ce os e>) - o es ^ 0000 r- es in r- ce es ¡ 1 t* y? 1 — -H °° S < os es ^ r^. ce es ce os r- to to , ce o t S a S î o -h fe es ça a a ■ Os ceg • 5 S al s g ; v - S i es o P OS 1 ■ iß ce t— ce 2^3ce »h« slfi r- es p -. ( 1 o oo t* O* ' — « co (O -*î" ce iß a. ce : 00 ce p 2 , 1 a iß K c» ^ os r- ■ es — ce o i iß« a O. I- —' h- es o ce es t* I ; r « . . iß o ■ m « 5 h_ es co OS CO l to M _ f. to 1 ■ os ce ! es as 00 iß ÎX es » ' T 00 1 ce o os 5s S ' >Ses ■ ; ce o < 1 >-« o 5 S « ÍSS|I § ß SS 1 es . 2 §s 5 ¡ : = "§§: o ce ^T* o ~ ' OS P g — os sa 1 © r* ^ ; °° « g 2 co — S?si os to tt co ; . O! Tf N V ;- [fl ce ' r- 2 00 o ce ce I OS —1 es . . iiî 00 ^" — -H CO *"* ■ en . es 1 ce iß 1 _ . __ s- os os os H<0 . os ■ t- o. . 72 IX ce io oj , ;-. ÇO ,rt iß eo « ^- » « «o — . > . 1- os es t- -H (O - 0© es »es — ■ ■ "ífS — guríes. , • • es ce r~ O — ■ -H CN M 1- ■ 1 os °-2 ' . ■ s a» - Soi co œ 1 ce ce .to Iß Iß 1- ■ g,,» N-1 iß — J¿T . »ß rt , -r es —. es . t- co ■ ci ■ —1 TT Ce ' "iß'M • ^* I 1 ce ce es iß ir3---rtrtißincNmrtrtcS' ce - ce ce ■ ■ Srtnc*»r-ißiß« «Nrt^ifiœN»roO-Nrt^iO»NMO)0«CNrt't"lÛ<ÛhX050'-iNCl5'ii/5!Or'900îO-HCNrtVii! cscNcsescsescscsesceeececeeecececeeece^^^^,fl',fl'^^^'fl"iß»ßißißißißißißißißtotototototo cNesesescscscsescsesescscsescscsesescNcscsesesescscscscsescscscscsescsescsescsescsescscsc^ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SUPPLEMENT S43 ce iß cs 00 tO t— ■vÎÊ ce*0 es'V *2__ co r~ — ço ' P o 2 c oo iß iß ce « es ce ce co cô5«H^ i OS t— 00 " ss3 ss ) f- r-ce ri ri f- OS CO > t- ce —i52 oo os i ce os ' «2 os S S P < io ce ■ £• h- iß ÇN —i r-» ! r- ce s S «s o co —■ce Tf CO ■* ee . £- i ot-- 00o OS iß ' -^ si S ■ '™| ce ^. « S S . o o o S §S ce 8 ! ! s SSôô J5 2 li- i ce ce « S S os to g ' ; ce« ce« ^ 88< I« HIß s? iZ °° — i os es os o —«CS t*. ' i"tO«' Slis iß os i iß iß £} to os w "5 co iß« x cs o oo csS21L es ^ os S es ^ • - ÇOi i m 2 - ^ ô ™ôo Vg o œ S to © -o i ■fl- os . osOS ■ OOoo ( ; í^ ée en ; ■ oo 2 to iß ""> 1 os ce ' 01 to ce °r ! «5es ^to ißoo ! "* SS $ v to ■ ' &■* S1"««« 1 OS OS i i2-S S2 w , - CO . OS os a 3i - 2 § E s I iß CO < iß ce . 1 cs oi So ■ o os i¡ E i p r- i es i^- "* ce ce ce ^ t ' « —feo es ce ! -g S ^ ' ■ ißt~ p . ■Si 1 t- o "« i ce ce ' S: os f-, i ce oo t i cs ce O M OS 'HH«| iß ce i *. . . „.eo^S^wS . ■ —, ■ V —! t- t~ CS ■ - -* n 9i i r- N rt n « Ol ^ es ce ce iß nißf-nMO>-,pH^rt«fu5-H' ce ce ^r iß iß i « t- — — ^H I t—I" -o6 U 3 9 i nt»"5 . iß ce p « es ffl «—•ce o ce ■ t*» ce t~ < s os os r- -- us -i « to ■*■ • os os t~ ce r^cecsTrescsosee—in t- iß es . ißnH . .ce. .—i. .iß-fl«. i ci ■ m ■ nt- ■ « n ■ t m -osee iceceeeißcs—,eeceesißißt-ceceh-^H -H-Heece^h-csiflcece iß ce ce iß i License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use S44 SUPPLEMENT es ce iß o > cs r- 00 : io tO r" : ¥ co to — r- ce i « SS00 — iß . —• os > o cs SI« 10 t- 1 os os , o — ; es t s S2 t- t- t- ¡ 1 ce tp i ce es i Í OS « Se ■ oo oo r- co < 5 oo — co iß *ß íS : • iß r- t ; oo s 23 - ; os es t- . iß t».os ess i' , x t* ' 111 ~* o ■ iß N ■ 5 os N M | liili ce ce =; S s *î= s s ce r*- O ÇO g . 00 tO i o?3 : 1 iß tp ÇO os 3g ce -v < :fc3 ■ os o ■ ce —« ' , OS 00 : s es »ß iß ao ■ p t-1 g **■ ~* > VtDi ¡1 tO o i p¡! Ilili es os ' sg! , Iß Iß i Tf t- ce r- ■ ■ to p ^ lß CO OS iß l ' 2 r» £ 0 co — t- 2 — ce <=»S S , 2 p oo« Iß r: oc to toIß , as p i 1 .*í V S ■ os r? i ao t— to os ( Smoí ce os Sí , 2g " h_ ce tr t- ao t— —■ CO ■ —« I- es Î2 os os SoSi: î^ ce o ce ce co es ' es ao ° S «5 « « es O « i | . iß Iß S cs to "5«Ojß^ S i ce cft ! tf S cs t- os ^ ce iß Cft t- S tf oo ! r; ' «ci cftQOcceeSLrißO' iß i iß ce ' i t: iß ao 5S es © . iß t>- -«T O to ■ SSE! •«f v ' es S2 cft èe t- ~+ r- o ooCNes'VfCi.cir-.i ^2 iß l'- —i o io es o ^> cs cs o 9S . tf g oo es t,c^^ . O ee os o < es ^ iß to ; 3 § osçp—lOasceißcoce co ce ig! OO 'V os ■ iß iß SCOo osiß co— ce oo i to ce tf ce ce r- te — »§§SSs tf Iß tf CO ce es o . os'SS'^SSceesto^ cs es — . ^" ^^ p m t- m x s ' §tofrt-[~ce^irtt-to CO -M f- Ex ^ ' oo cs o ao * *ß ÎZ iß o c o to g o i t— — os cc ' s oiSS^^ooSooosißce ao £r to ' iß ce —" o oooOcftSptftfTft- ce£ osS3 1 . o °° II t^ S2 ; ^ s2 ' iß O iß OS r- f^ es —. mVf-Qc5-'(íiNiNrt ; - § cs - o5 i Er S es ao i , os o OS — o cs o cs , , ce t— ■ ^►i. oíí£ S=; , 1 oo oo , CO iß 00 'H N N ! ce iß , ' t- cs to —• Cft ce ; oc iß —■ . OO Iß i -* es »ß 00 ! Silto to i S2=2¡ ¡gagas' S i : os » 35 iß £: es ce ■ w t- ^ - ! to OS ' < —i f1- s 2 3 ,co«; i — S . J totO r,f- aos ¡5 ißS-- -, tr :?22sseses « 3 ; ï °° 2 < v ■ oso _ « At*" ! 2 « ce ; t: iß • o cs os ce es t— ; 5 " S 3» es ce ce i -^ co 2«S2 es to ce os . ■ to es es ^" ; •es . „ 0 — iß co _ — — iß t— ce ' • (^ f- t^ to es -, cs ce £ os iß o ' iß o i ) os ;J ce ce -h ce -H 00 i OO —• < ifs ■ es ^ ce , 1 w ce ■ • ¡ ! « t- N i« "* ' ¡ cft es r- —< t— I -H os . I o os - 'ß • '-i co es ai ce o « • c* i t— iß ce ce 1 iß ce ce iß ■ > —< ^- iß tf i ■ — — ce ce i tfißceeeeet— cs^-,- ce ce —t iß t-- cs ce License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SUPPLEMENT S45 • ce as iß o os t- ce cs t- —i es ce co f- S ce o1 o ■ cß _ 'r. CS f. ap ^ ce "^ as ce os S 2g ce co iß O. Iß Tf OS t- Tf O t- iß Oto ^ - to ee iß oo r- Tf ce os ■* ; ce rr -2 ce S 2! ' ¡s os iß co ce oo oo ce oo iß t—iß to ee ^ s co to ce oo tp H 3. iß iß to es S oo ce P£ ce iß o es o to as r- Jï to §rsg?ss 00 tf es 00 H kß I Iß P os N "^ t-. to Sg tO v IS Scs V to ^Ca. °°o [ S? Ti —i oo t~ Tf os to ■ iß OS « TP cs —, tO ce os £ es . o to oo SK§ . o os cs —i CO oo ; to to Tf S - * -» 22 to -< cs S î 00 t- CO »ßSi tO . Tf ce t- : £• ee —« ^ -h t; ~* ce î os es Tf • ?ls SS2| o, p es - os o ; cs os — S§2 -,. wS ce** S Iß t- ; ■ oo tp to o p : os S -h5sl~S§ >■*—i ce es OS (0 2se ^ " ce£ » EH QO ■ CO Tf xß ( i§ s. o 2 os î =:g S§! cm a - . cft I COÏt8 is. * rt■ ! o — 'S TP ee esS oo o ao ^_ ee oo S t- ce to ,; ï Os ee . Iß OO tO L" r^ Tf f- os ■ t- es iß xv es tf iß ce îr oo yce assi* oo": totf tor- osen - a. — »o ; co cs • •j gX as iß _. .ißIß . tO Tf , —■ to •"-" cftS2 o2 cft2 tpy, ™es ■ , ^ oo ce * o to o OS tt Tf « 2 « ¡ß os 12 ce os es t- «CN° -, 00 (» Tf î r- co iß ° £ S °°2«| ? ? -r ssS es — to TP O < to *e !-S O S "2 os —. £r i 1 to CO —H k, tO K, o " oo . to g ce Iß U) » I ' o a os 2 os ee S t- —it; ^Tf • tp t- "^ t- ce ***ïS _ nr> os_ esir\> cei-i^ ce •""' ■ cs ■ 00 OS —i gx r- iß • o ee ' ce - ce S ce to to es co i »"si —' os to t- t- I 'ïrSï^S.S^S Siß r- O/1 os ee as£î «„ t*- to iß Sä O55« 3 . Tf Iß o ce i ■ iß os 22 j to Cft co —*o o . 1 iß T* ,_, Tf ce es CT, . os ce i * es -, ^ CS iß . es o ««. — t* f- I 3 O tf ce f- t- co ce i i "~l es o h-PSOci i ce tp co t- ■ iß iß ' 1 os ■ tp es t- ' i • o o -» -2 t- —i ce ce iß ce iß ce tf w* i r- cs ce os os " i O) h ■ et i iß r- iß ■-" ce i i H(OMMI o,eece—'ißißaoce -HcsceTPißcot— oooso—icsceTTiotot^coaio—•csceTpißcot-cooso—'CseeTPißto r— cooso-^escetp escsesescscscscsescecececeeececeeeeecetptfTPTPTfTftptftpTfißißißißißißiß ißißißtocotococo cscsescscscscscscscscNCscscscscsesescscscscsesescscsescscscscscscscscscs esescscscscsescs S î- 2 ' ee 2 t- « 2 o s 1 00 : to § t- >—1 ï es g iß Cft 00 iß 9Bg œ P to ce P 00 es os I S TP CO ^ t- r- ci £ 01 - os çe o iß S S« CO TÍ S ï o iß o f~2! oses i co ^ = i«; . 2 ce co f- CN 1—1 1 S 2 S f- es g iß ce . a M os 5 CM t© œ. S License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use S46 SUPPLEMENT ee f^J io iß X, f- ce "" co es ' o o os to ce o Os r- ~- iß —, os ce t- es ce « tp ce ee co t- iß go ■ to iß to os Tf O ce o os co o es ce tp —,O ceIß eeTf » çe ce a OS iß Iß CS CO OS CS f- o tO O aes S p OO iß to os ce ee co «ICO iß es iß os »Sceg ce os os a, Tf os os tp 7Ïa i«es «2<-" as — as iß p HlßlßS to iß ee to rt io tr ,ï iß os os »¿i r- tp io ce ao 'S S g es 5 10 OS tf if> cs oo t- es ■ o. 3£2g os os to I os - t- n ~ -, es S- Tf Tf O 3? os iß » ao co K es o Tf es 5 s <° S ■ « 2 OS cs §g2g 18 2 ce h: t- es 28S1 c t es t- ee cSg 2£ II fcg t-o ceo •°5g Iß to s a. iß 2 w « î?5 aO ißos oiß ff -2 t-cg S g çs ee S cs o ce p m S » to os S N x S 1 <5 ce 8Sg| i8 s tf es çe . > iß h. £i icsißceeeißr-as—icece l«HH«rtMßHH I —i iß iß ee ce mtONXOOHMrtVißCNXCIlOHCSrtTißtDr-XOlO-HiNrtV iß CO t— OO OS O (ÛtOtOtOtOr-t-Nr-t-Nr-r-t-t-XXXXXXXXXœOiOSOSOSa OS O) OS O) OS O cscsesescscsescscsescscscscscscscscscscscsescsescsescscscscs escsescsesee License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use