Factorizations of Cunningham numbers with bases 13 to 99

arXiv:1004.3169v2 [math.NT] 20 Apr 2010 Factorizations of Cunningham numbers with bases 13 to 99: Millennium edition∗† Richard P. Brent Oxford University Computing Laboratory Wolfson Building, Parks Road Oxford OX1 3QD, UK rpb@comlab.ox.ac.uk Peter L. Montgomery 780 Las Colindas Road San Rafael, CA 94903–2346 USA pmontgom@cwi.nl Herman J. J. te Riele Centrum voor Wiskunde en Informatica Kruislaan 413 1098 SJ Amsterdam The Netherlands herman@cwi.nl with the assistance of Henk Boender, Stephania Cavallar, Conrad Curry, Bruce Dodson, Jens Franke, Joseph Leherbauer, George Sassoon and Robert Silverman PRG TR-14-00 31 December 2000 Abstract This Report updates the tables of factorizations of an±1 for 13 ≤a < 100, previously published as CWI Report NM-R9212 (June 1992) and updated in CWI Report NM-R9419 (Update 1, September 1994) and CWI Report NM-R9609 (Update 2, March 1996). A total of 951 new entries in the tables are given here. The factorizations are now complete for n < 76, and there are no composite cofactors smaller than 10102. This “Millennium edition” gives the complete tables incorporating all updates. A file containing only the new updates, a file containing factorizations for an extended table range, and a file of factors, are all available on the internet. 1991 Mathematics Subject Classification. Primary 11A25; Secondary 11-04 Key words and phrases. Cunningham numbers, elliptic curve method, factor tables, number field sieve, quadratic sieve, ECM, GNFS, MPQS, NFS, PMPQS, PPMPQS, SNFS ∗Incorporating Factorizations of an ± 1, 13 ≤a < 100: Update 3. This version corrected 18 March 2001. †Copyright c⃝2000, the authors. rpb200 typeset using LATEX 1 Introduction For many years there has been an interest in the prime factors of numbers of the form an ± 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with an elements, and the multiplicative group of F has an −1 elements. Also, for prime a the sum of divisors of an is σ(an) = (an+1 −1)/(a −1). Numbers of the form an + 1 arise as factors of a2n −1 and in other ways. An extensive table of factors of an ± 1 for a ≤12 has been published by Brillhart et al [11]. The computation of these tables is referred to as the Cunningham Project in recognition of the pioneering computations of Cunningham and Woodall [12]. For a history, see the Introduction in [11]. The tables [11] are limited to a ≤12, but many applications require larger bases. In June 1992 tables covering the range 13 ≤a < 100 were published [8]. The exponents n satisfied an < 10255 if a < 30, and n ≤100 if a ≥30. An update [9] containing 780 new factorizations (with the same limits for a and n) was published in September 1994, and a second update [10] containing 760 new factorizations was published in March 1996. These factorizations are now incorporated in the Magma package [3]. Since the second update [10], many new factors have been found. The factorizations are now complete for n ≤75, and there are no composite cofactors with fewer than 103 digits1. This report includes all the new (complete or partial) factorizations found from the publication of [10] to 31 December 2000. Altogether, 951 new (complete or partial) factorizations are listed, involving 1098 new factors2. Table 1 summarizes progress since the publication of the original tables [8]. “Update 3” refers to this Report. Table 1: Statistics regarding the Tables and Updates Tables Date Smallest Complete to Total composite exponent entries Original June 1992 81 digits 46 13882 Update 1 Sept. 1994 87 digits 58 780 Update 2 March 1996 95 digits 66 760 Update 3 Dec. 2000 103 digits 75 951 Table 2 shows the number of prime factors of different sizes found for Updates 1–3 (excluding large factors obtained by division). The median sizes are 26 digits for Update 1, 29–30 digits for Update 2, and 33 digits for Update 3. The smallest new factor is 20 digits for Update 3 (compare 14 digits for Update 2). We would be surprised if many factors of less than 25 digits are still to be found. The largest new penultimate factor is 60 digits for Update 3 (compare 56 digits for Update 2). Table 2: Distribution of Factors Digits Update 1 Update 2 Update 3 10–14 0 1 0 15–19 17 23 0 20–24 333 144 24 25–29 329 273 242 30–34 154 197 322 35–39 72 99 181 40–44 44 89 134 45–49 9 39 107 50–54 0 14 63 55–59 1 3 15 60–64 0 0 10 Total 959 882 1098 1“digits” always means “decimal digits”. 2Here and elsewhere we do not count large factors which are obtained by division by other factors. ii 2 Format of the Tables The format of the Tables is the same as in [8]. For each base a, not a perfect power, in the range 13 ≤a < 100, we give two separate tables – Table a−: factorizations of an −1, n odd. Table a+: factorizations of an + 1. The exponent ranges are as in [8] – 13 ≤a < 30, exponents n such that an < 10255. 30 ≤a < 100, exponents n ≤100. The entries are similar in format to tho

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