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 a^{n}±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 10^{102}.

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 a^{n} ± 1, 13 ≤ a < 100: Update 3. This version corrected 18 March 2001.

†Copyright \begin{matrix}c \\ ◯ \end{matrix} 2000, the authors.  rpb200 typeset using L^{A}TEX

1 Introduction

For many years there has been an interest in the prime factors of numbers of the form a^{n} ± 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 a^{n} elements, and the multiplicative group of F has a^{n} − 1 elements. Also, for prime a the sum of divisors of a^{n} is σ(a^{n}) = (a^{n+1} − 1)/(a − 1). Numbers of the form a^{n} + 1 arise as factors of a^{2n} − 1 and in other ways.

An extensive table of factors of a^{n} ± 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 a^{n} < 10^{255} 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 digits^{1}. 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 factors^{2}. 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.

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 a^{n} − 1, n odd.

Table a+: factorizations of a^{n} + 1.

The exponent ranges are as in [8] –

13 ≤ a < 30, exponents n such that a^{n} < 10^{255}.

30 ≤ a < 100, exponents n ≤ 100.

The entries are similar in format to those of the “short” tables in [11]. All known factors, including algebraic and Aurifeuillian [4] factors, are listed. Factors which are given as decimal numbers are primes. Exponents are indicated by a hat (ˆ ), for example “2ˆ 3” means 2^{3}. Multiplication is indicated by a period (.), for example 3^{3} + 1 = 2^{2}7 is written as “2ˆ 2.7”. A period at the end of a line implies that the factorization is continued on the next line. An underscore ( ) at the end of a line means that a (large) factor is continued on the next line (see, for example, the entry for 19^{177} − 1).

The largest factor of a^{n} ± 1 may be found by division by the smaller factors. Thus, such factors are abbreviated. The notation p_{xy} or “pxy” means a prime factor of xy digits. For example, the prime 1238926361552897 might be abbreviated as p16. Similarly, the notation c_{xy} or “cxy” means a composite number of xy digits.

3 Availability of Tables, Updates and Factors

The complete tables for the original table range are given below. The changes since Update 2 are available

by anonymous ftp from ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/

rpb134u3.txt.gz (a compressed text file). This file includes comments on the person and method responsible for finding each factor (if there is no attribution, the factor was found by one of the authors).

The complete tables incorporating Updates 1–3, and a list of factors, are also available online: see

http://www.comlab.ox.ac.uk/oucl/work/richard.brent/factors.html The restriction n ≤ 100 for

bases a ≥ 30 has been relaxed for the online tables; we only require a^{n} < 10^{255}. For this extended table range, the smallest composite has 102 digits.

4 Factorization Methods

Since Update 2 we have attempted to factor the remaining composite numbers in the tables by using the elliptic curve method (ECM). Sometimes ECM is successful in finding one or more factors. If the factorization can not be completed by ECM, but the remaining composite part is sufficiently small, we use the multiple polynomial quadratic sieve (MPQS) method to complete the factorization. In some cases we prefer to use the number field sieve (NFS) if it is predicted to be faster than MPQS^{3}.

We do not describe ECM, MPQS or NFS here. The reader should refer to [16, 17, 19] for a general description of ECM, to [2, 23] for MPQS, and to [15, 13, 21] for NFS. A recent survey is [7]. The particular implementations of ECM by Brent and Montgomery are described in [6, 18].

Table 3 shows the number of factors found by several methods in the preparation of Updates 1–3. For ECM and MPQS these only include penultimate factors of at least 30 digits. An increase in the use of SNFS and decline in the use of Pollard’s p ± 1 methods [22] is evident. There is also a marked increase in the number of large (at least 30-digit) factors found by ECM. Most of the new factors found by MPQS and SNFS are large because these methods are only used after ECM has been tried. In fact, since Update 2, MPQS and SNFS did not find any factor with less than thirty digits, because such factors had already been found by ECM. The largest factor found by ECM was a 52-digit factor of 96^{98} + 1 (see [5]).

3The choice depends upon the size of the known non-algebraic factors of the number a^{n} ± 1. We normally use the special number field sieve (SNFS), but in at least one case (17^{186} + 1) the general number field sieve (GNFS) was used (this factorization was contribution by Couveignes, Granboulan, Hoogvorst and Nguyen).

Table 3: Factors Found by Different Methods

Pollard Pollard ECM MPQS NFS
p − 1  p + 1 (30D+) (30D+)
Update 1  38  16  69  157  37
Update 2  0  3  151  155  136
Update 3  0  3  423  129  279

5 First Holes

A “first hole” is the first composite number occurring in a table. Thus, each table of factorizations is complete up to, but not including, its first hole. Table 4 lists the exponents of the current first holes for 2 ≤ a < 100 (the range 2 ≤ a ≤ 12 is included for the sake of comparison). For example, the first holes in the tables for a = 17 occur for exponents 137 and 118. In fact, first holes such as 17^{118} + 1 = 2 · 5 · 29 · 7789 · c_{139} are good candidates for factorization by SNFS.

Table 4: Exponents of First Holes for 2 ≤ a ≤ 99

a − + a − + a − + a − +
2 641 617 3 379 382 5 307 283 6 251 232
7 227 214 10 197 223 11 191 181 12 173 172
13 161 151 14 149 134 15 127 122 17 137 118
18 131 121 19 155 113 20 149 106 21 125 128
22 103 116 23 101 101 24 101 107 26 107 103
28 103 106 29 101 112 30 103 103 31 97 113
33 103 89 34 103 101 35 97 103 37 89 97
38 101 86 39 89 89 40 97 97 41 101 89
42 115 86 43 101 89 44 103 94 45 83 92
46 101 82 47 89 86 48 107 94 50 89 97
51 97 83 52 83 82 53 89 88 54 107 79
55 107 86 56 83 79 57 83 79 58 83 76
59 97 79 60 79 86 61 79 94 62 79 82
63 83 83 65 79 79 66 97 86 67 83 82
68 79 76 69 83 83 70 83 89 71 89 76
72 79 83 73 79 83 74 89 82 75 79 79
76 103 79 77 89 86 78 79 79 79 97 83
80 83 82 82 83 79 83 79 82 84 79 76
85 83 76 86 79 79 87 97 83 88 79 92
89 83 83 90 83 79 91 83 92 92 83 82
93 79 82 94 79 76 95 79 79 96 83 79
97 83 82 98 95 79 99 89 88

6 Probable Primes

Numbers listed as prime have not in all cases been rigorously proved to be prime; they may merely have passed a probabilistic primality test [14]. There is a positive but extremely small probability that a composite number will pass such a test and be mistaken for a prime. In applications where it is essential for primality to be proven rigorously, one should apply an algorithm such as Morain’s elliptic curve primality test [1, 20], which can easily prove or disprove the primality of numbers of the size considered here.

Acknowledgements

We thank the people mentioned on the title page for their assistance since Update 2. In particular, Robert Silverman helped eliminate many first holes.

Thanks to Franz-Dieter Berger, Anders Björn, Henk Boender, Arjen Bota, Lars Brunjes, John Cannon, Stephania Cavallar, Graeme Cohen, Conrad Curry, Nicolas Daminelli, Thomas Denny, Bruce Dodson, Harvey Dubner, Robert Dubner, Will Edgington, Jens Franke, Torbjörn Granlund, Tom Hill, Uwe Hollerbach, Rob Hooft, Marije Huizing, Wilfrid Keller, Yuji Kida, Hidenori Kuwakado, Samuli Larvala, Joseph Leherbauer, Arjen Lenstra, Paul Leyland, Walter Lioen, Hisanori Mishima, Peter Moore, Mitsuo Morimoto, Andreas Müller, Henrik Olsen, John Pollard, Hans Riesel, David Rusin, George Sassoon (Mullfac), Robert Silverman, Thomas Sosnowski, Allan Steel, Andrew Steward, Samuel Wagstaff, Georg Wambach, Damian Weber, Michael Wiener, Hugh Williams, Aiichi Yamasaki, Paul Zimmermann, those mentioned in a previous footnote, and anyone inadvertently omitted from this list, for sending us factors or assisting in other ways.

The Australian National University Supercomputer Facility provided computer time to run the first author’s ECM programs on a Fujitsu VP 2200/10 vector processor and an SGI Power Challenge. The ANU-Fujitsu CAP Research Project provided time on a Fujitsu AP 1000.

The Oxford Supercomputing Centre provided time to run the first author’s ECM programs on Oscar, an SGI Origin 2000.

The Dutch National Computing Facilities Foundation provided computer time to run the programs of Boender, Elkenbracht-Huizing, Montgomery, and te Riele.

Many runs with ECM, MPQS and SNFS were carried out on workstations at CWI and Leiden University. We are grateful to the workstation “owners” at CWI and Leiden University for letting us use their idle cycles for this project.

References

[1] A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp. 61 (1993), 29–68.

[2] H. Boender and H. J. J. te Riele, Factoring integers with large prime variations of the quadratic sieve, Report NM-R9513, Department of Numerical Mathematics, Centrum voor Wiskunde en Informatica,

    Amsterdam, July 1995. ftp://ftp.cwi.nl/pub/CWIreports/NW/NM-R9513.ps.Z .

[3] W. Bosma and J. Cannon, Handbook of Magma Functions, School of Mathematical Sciences, Univ. of Sydney, January 1996. See also the Magma home page: http://www.maths.usyd.edu.au/comp/

    magma/Overview.html .

[4] R. P. Brent, On computing factors of cyclotomic polynomials, Math. Comp. 61 (1993), 131–149.

[5] R. P. Brent, Large factors found by ECM, Computer Sciences Laboratory, Australian National Univ., Canberra, December 1995. Revision available from ftp://ftp.comlab.ox.ac.uk/pub/Documents/

    techpapers/Richard.Brent/champs.txt

[6] R. P. Brent, Factorization of the tenth Fermat number, Mathematics of Computation, 68 (1999), 429-451.

[7] R. P. Brent, Recent progress and prospects for integer factorisation algorithms, Computing and Combinatorics: Proc. Sixth Annual International Computing and Combinatorics Conference (Sydney, July 26-28, 2000), edited by D.-Z. Du, P. Eades, V. Estivill-Castro, X. Lin and A. Sharma, Lecture Notes in Computer Science, Vol. 1858, Springer-Verlag, Berlin, 2000, 3-22. Preliminary

    version available at ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/
    rpb196tr.ps.gz

[8] R. P. Brent and H. J. J. te Riele, Factorizations of a^{n} ± 1, 13 ≤ a < 100, Report NM-R9212, Department of Numerical Mathematics, Centrum voor Wiskunde en Informatica, Amsterdam,

    June 1992, v+363pp. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/
    rpb134.*.gz

[9] R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Update 1 to: Factorizations of a^{n} ± 1, 13 ≤ a < 100, Report NM-R9419, Centrum voor Wiskunde en Informatica, Amsterdam, Septem-

ber 1994, iv+42pp. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/
rpb134u1.*.gz

[10] R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations of a^{n}±1, 13 ≤ a < 100: Update 2, Report NM-R9609, CWI, Amsterdam, March 1996, viii+42pp. ftp://ftp.comlab.ox.ac.uk/pub/

Documents/techpapers/Richard.Brent/rpb134u2.*.gz

[11] J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b^{n} ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI, 1988.

Also various updates: see http://www.cerias.purdue.edu/homes/ssw/cun/ .

[12] A. J. C. Cunningham and H. J. Woodall, Factorisation of y^{n} ∓ 1, y = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers (n), Hodgson, London, 1925.

[13] R. M. Huizing, An implementation of the number field sieve, Experimental Mathematics 5 (1996), 231–253.

[14] D. E. Knuth, The art of computer programming, Volume 2: Seminumerical algorithms (third edition), Addison-Wesley, Menlo Park, CA, 1998.

[15] A. K. Lenstra and H. W. Lenstra, Jr. (editors), The development of the number field sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, Berlin, 1993.

[16] H. W. Lenstra, Jr., Factoring integers with elliptic curves, Annals of Mathematics (2) 126 (1987), 649–673.

[17] P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp. 48 (1987), 243–264.

[18] P. L. Montgomery, An FFT extension of the elliptic curve method of factorization, Ph. D. dissertation, Mathematics, University of California at Los Angeles, 1992.

[19] P. L. Montgomery, A survey of modern integer factorization algorithms, CWI Quarterly 7 (1994), 337–366.

[20] F. Morain, Courbes elliptiques et tests de primalité, Ph. D. thesis, Université Claude Bernard – Lyon I, France, 1990.

[21] B. Murphy, Polynomial selection for the number field sieve integer factorisation algorithm, Ph. D. thesis, Australian National University, Canberra, 1999.

[22] H. Riesel, Prime numbers and computer methods for factorization, 2nd edition, Birkhäuser, Boston, 1994.

[23] R. D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), 329–339.

Part II – The Tables

There have been many improvements since the original tables were published in June 1992. The original tables included 3093 composites (i.e. incomplete factorizations); this number has been reduced to 1027. Following are the complete updated tables with the same range as the original table. See §3 for information regarding online versions of these tables (with an extended table range) and associated factors.

                          Table 13-
13   1-  2^2.3
13   3-  2^2.3^2.61
13   5-  2^2.3.p5
13   7-  2^2.3.p7
13   9-  2^2.3^3.61.p7
13  11-  2^2.3.23.419.859.p5
13  13-  2^2.3.53.264031.p7
13  15-  2^2.3^2.61.4651.30941.p6
13  17-  2^2.3.103.443.p14
13  19-  2^2.3.12865927.p13
13  21-  2^2.3^2.43.61.337.547.2714377.p7
13  23-  2^2.3.1381.p22
13  25-  2^2.3.701.9851.30941.p16
13  27-  2^2.3^4.61.650971.1609669.p14
13  29-  2^2.3.1973.2843.3539.p21
13  31-  2^2.3.311.1117.p28
13  33-  2^2.3^2.23.61.419.859.18041.p23
13  35-  2^2.3.211.30941.5229043.3357897971.p15
13  37-  2^2.3.1481.67495678093.4287755796749.p14
13  39-  2^2.3^2.53.61.79.1093.4603.21841.264031.1803647.p14
13  41-  2^2.3.6740847065723.p32
13  43-  2^2.3.119627.p42
13  45-  2^2.3^3.61.181.4651.30941.161971.1609669.p25
13  47-  2^2.3.183959.19216136497.p36
13  49-  2^2.3.1667.5229043.28082195177.p34
13  51-  2^2.3^2.61.103.443.763879.15798461357509.p30
13  53-  2^2.3.107.194723.189541180943969.8403659652641423.p21
13  55-  2^2.3.23.419.859.2861.18041.30941.13545148572117361.p25
13  57-  2^2.3^2.61.12865927.796956375829.9468940004449.p29
13  59-  2^2.3.273997.5311771.p53
13  61-  2^2.3.4027.4759.7687.27817.92110001.4672993939.
         48401662036451.p20
13  63-  2^2.3^3.43.61.127.337.547.6301.825679.1609669.2714377.
         5229043.7327657.997294663.p13
13  65-  2^2.3.53.131.1171.30941.156131.264031.1803647.
         71442881968439190301.p24
13  67-  2^2.3.586079017.1093561021297.1195860242597359.p38
13  69-  2^2.3^2.61.139.1381.10903.282244620282733.
         2519545342349331183143.p29
13  71-  2^2.3.6959.12923.201499.p65
13  73-  2^2.3.145009586102490829218552548223336637.p46
13  75-  2^2.3^2.61.701.1951.4651.9851.30941.161971.
         2752135920929651.p42
13  77-  2^2.3.23.419.859.18041.5229043.624958606550654822293.p47
13  79-  2^2.3.3793.16433.6709792556882923.p64
13  81-  2^2.3^5.61.650971.1609669.57583418699431.
         1102123844336048491.p42
13  83-  2^2.3.12451.1383113.p82
13  85-  2^2.3.103.443.30941.28152511.15798461357509.
         11435433293542010176161611.p39
13  87-  2^2.3^2.61.1973.2843.3539.8527.846041103974872866961.
         808648601294417626878199.p35
13  89-  2^2.3.179.9257.716164560927079240026189379811.p62
13  91-  2^2.3.53.10193.34763.264031.1326781.1803647.5229043.
         72019220497491955875703699.p40

13  93-  2^2.3^2.61.311.1117.1303.6377362657.109711631401199502223.
         8170509011431363408568150369.p34
13  95-  2^2.3.191.27361.30941.4986361.12865927.9468940004449.p67
13  97-  2^2.3.389.971.93964390627.p91
13  99-  2^2.3^3.23.61.199.419.859.3169.18041.1609669.
         17551032119981679046729.2192746830056885246381227.p37
13 101-  2^2.3.1213.p109
13 103-  2^2.3.5563.9577825183.299872566772439874463.
         1493185475735966076741431.p56
13 105-  2^2.3^2.43.61.211.337.547.4651.30941.161971.2714377.
         5229043.3357897971.995277238201.707179356161321.p42
13 107-  2^2.3.2141.3286126349184953.p100
13 109-  2^2.3.1091.2617.794018567.7730096009.p96
13 111-  2^2.3^2.61.1481.67495678093.4287755796749.31964044249933.
         5897148699869174859194378551.p53
13 113-  2^2.3.1583.5651.114025363.p110
13 115-  2^2.3.1381.30941.3344431.1309485451.2519545342349331183143.p83
13 117-  2^2.3^3.53.61.79.1093.4603.21841.259507.264031.1519831.
         1609669.1803647.19440694027.57745124662681.
         462116939124761758477693.p35
13 119-  2^2.3.103.443.5229043.13883731.15798461357509.
         1251321793446942324872434004484031.p67
13 121-  2^2.3.23.419.859.18041.4911149.4161245903.1368405847627633.p92
13 123-  2^2.3^2.61.1340947.6740847065723.1374332707191583.
         763604898951326591407.34241383511160276465403.
         2500231913819762291766481.p32
13 125-  2^2.3.251.701.9851.30941.350101751.801560380001.
         2752135920929651.p89
13 127-  2^2.3.5912867.190421635185941.115806724907473978817.p100
13 129-  2^2.3^2.61.6451.119627.7180925126477041.43466212635494347.
         552781743698966779174737704265497702530829.p58
13 131-  2^2.3.263.135979.400852141.p129
13 133-  2^2.3.2129.5229043.12865927.9468940004449.p117
13 135-  2^2.3^4.61.181.2161.4651.30941.161971.203311.650971.
         1609669.57583418699431.2997534286720771645924621.p72
13 137-  2^2.3.p152
13 139-  2^2.3.62895574665131573770495625630815229.p119
13 141-  2^2.3^2.61.283.1693.183959.1924651.19216136497.
         455036140638637.534280344481909234853671069326391741.p76
13 143-  2^2.3.23.53.419.859.18041.264031.1803647.133419712999.
         140748861026489.34436567931553789.39826785573214737241.
         422385642452919347685571782874181.p40
13 145-  2^2.3.1973.2843.3539.30941.846041103974872866961.
         3331836160337746180840402301.
         573409492728603405512028102226141331.p62
13 147-  2^2.3^2.43.61.337.547.1667.2714377.5229043.28082195177.
         1303929619014323295195519453212113.p94
13 149-  2^2.3.2242046914771.p153
13 151-  2^2.3.2118633909226902785023835895222977.p134
13 153-  2^2.3^3.61.103.443.919.554167.763879.1609669.326881441.
         1814824801.15798461357509.1981729131748453.
         535301324317729848039542278039.p66
13 155-  2^2.3.311.1117.30941.100860853831.46458503064581.
         100889415553845820735921.8170509011431363408568150369.p86
13 157-  2^2.3.80071.27323390907418173038175977.p144
13 159-  2^2.3^2.61.107.19081.194723.189541180943969.
         8403659652641423.274690322498001718403.p112

13 161-  2^2.3.1381.3221.5229043.2519545342349331183143.c144
13 163-  2^2.3.8693117.1279033291.57483449583854824247.p145
13 165-  2^2.3^2.23.61.419.859.2861.4651.18041.30941.161971.
13545148572117361.17551032119981679046729.
8603427927040724342317061.
3386047926964307416018724773840321.p56
13 167-  2^2.3.74016071.199494205465879.863628686858502013448191.c139
13 169-  2^2.3.53.677.264031.1803647.3221968439.
200764162484293733261.586814952741629335333029441607.
9678439913990009887548497710458137.p78
13 171-  2^2.3^3.61.1609669.10692289.12865927.796956375829.
9468940004449.1707914176047127.
11579626908585447176195979823.
14654187650846568795703728901.p70
13 173-  2^2.3.347.113489.481633.p179
13 175-  2^2.3.211.701.9851.30941.667801.5229043.3357897971.
6960114001.707179356161321.2752135920929651.
15320444013418579351.97940332989443462362601.
1534984704054820657287614317690451.p43
13 177-  2^2.3^2.61.1063.273997.5311771.274985077.
210439555242381217.39514351240970968031506068648339967.
30233706222041559662776806806113588758534692993333729.p66
13 179-  2^2.3.68014208993.p188
13 181-  2^2.3.5431.23531.30047.423541.32598463.
1244504930700295881445351.c151
13 183-  2^2.3^2.61^2.367.4027.4759.7687.27817.92110001.4672993939.
6274983367069.48401662036451.71639002544177046079.
87103671046409436847.417324063726420174777931.
715867003033745576109669960047077.p41
13 185-  2^2.3.1481.30941.30594191.67495678093.355505024671.
4287755796749.6934871644321.26101416485231.31964044249933.p116
13 187-  2^2.3.23.103.419.443.859.18041.28123679.15798461357509.
445536601753454704112226421.p145
13 189-  2^2.3^4.43.61.127.337.547.757.6301.650971.825679.1609669.
2714377.5229043.7327657.997294663.203865063949.
2617880963923.57583418699431.91323298699395463.
56199108572218966663.p70
13 191-  2^2.3.3674491999.34115367574721.
84897381032489674968541203989617.p157
13 193-  2^2.3.27281323.223486667.1468937560453708434469.c177
13 195-  2^2.3^2.53.61.79.131.1093.1171.4603.4651.21841.30941.
156131.161971.264031.1302991.1803647.8078071.56253601.
743408641.57745124662681.5234176303106791.
71442881968439190301.158943831041162255277151.
2794855972087224486028011777601.p32
13 197-  2^2.3.81559.c214
13 199-  2^2.3.797.5573.94327.1562947.21460559.60144579533.
2822026772941.704550879887947.c158
13 201-  2^2.3^2.61.1609.586079017.1093561021297.1195860242597359.
14137511716656949.46832111174371450604616850217838043873.c128
13 203-  2^2.3.1973.2843.3539.82013.1050323.5229043.11878648531.
615557670589.846041103974872866961.p155
13 205-  2^2.3.1231.3691.30941.6740847065723.
143207110374732650171977021.
58047259977349384372529747126587.p146
13 207-  2^2.3^3.61.139.1381.10903.1609669.39674449.1598219928127.
282244620282733.2519545342349331183143.

22271867346104672186904295489.p128
13 209-  2^2.3.23.419.859.18041.12865927.50792017.864962803.
50273133791.9468940004449.c174
13 211-  2^2.3.318872063.453216197376414128940399015272903.c193
13 213-  2^2.3^2.61.6959.12923.201499.8227339.39355159.5990431171.
24385215615829.34078015035971503.
504649541791910254264659793435883391598383106891.
1016705137474747135476535700217615993757244294651630647.p65
13 215-  2^2.3.30941.119627.1395397213914606731.
552781743698966779174737704265497702530829.c169
13 217-  2^2.3.311.1117.16493.5229043.436265047.506813917.
8170509011431363408568150369.c179
13 219-  2^2.3^2.61.439.1024921.1762236499.
145009586102490829218552548223336637.
1194776740722409996051132396456283691052300333.c143
13 221-  2^2.3.53.103.443.77351.264031.1803647.710581301.3890188303.
15798461357509.
7720293918867096427863843822346904850820471837355601107472627048_
71450490130127391654067619897.
p98
13 223-  2^2.3.6691.883973.609240789000934129655613721.c211
13 225-  2^2.3^3.61.181.701.1951.4651.9851.30941.161971.1609669.
1711042515451.2752135920929651.2997534286720771645924621.
185129510660424877123166434146327362331151.c122
13 227-  2^2.3.362293.433117.c241

                          Table 13+
13   1+  2.7
13   2+  2.5.17
13   3+  2.7.157
13   4+  2.p5
13   5+  2.7.11.2411
13   6+  2.5.17.p5
13   7+  2.7^2.29.p5
13   8+  2.p9
13   9+  2.7.19.157.271.937
13  10+  2.5^2.17.421.601.641
13  11+  2.7.p12
13  12+  2.14281.p9
13  13+  2.7.13417.20333.p5
13  14+  2.5.17.p14
13  15+  2.7.11.31.157.2411.p8
13  16+  2.2657.441281.p9
13  17+  2.7.p18
13  18+  2.5.17.37.28393.428041.p7
13  19+  2.7.p21
13  20+  2.41.14281.29881.p12
13  21+  2.7^2.29.157.463.22079.p11
13  22+  2.5.17.5281.p19
13  23+  2.7.47.277.1151.2347.p14
13  24+  2.1009.407865361.p15
13  25+  2.7.11.101.2411.57751.p16
13  26+  2.5.17.380329.p22
13  27+  2.7.19.157.163.271.937.904663.p12
13  28+  2.113.14281.p25
13  29+  2.7.59.1741.8546789918171.p14
13  30+  2.5^2.17.421.601.641.28393.460655521.p10
13  31+  2.7.373.2729.145831193.p20
13  32+  2.193.1601.10433.p26
13  33+  2.7.67.157.331.1123.6997.122167.960961.p12
13  34+  2.5.17^2.1021.897329.61165661.75094577.p10
13  35+  2.7^2.11.29.71.2411.22079.654221.759641.p14
13  36+  2.73.4177.14281.181297.815702161.p16
13  37+  2.7.223.21017.152219.1548921490187.p17
13  38+  2.5.17.229.94621.p33
13  39+  2.7.157.13417.20333.79301.p27
13  40+  2.407865361.p36
13  41+  2.7.83.638453.140299545168523469.p20
13  42+  2.5.17.673.2857.4621.28393.23161037562937.p17
13  43+  2.7.173.12566837.8001003293.346982008721.p16
13  44+  2.89.6073.14281.p39
13  45+  2.7.11.19.31.157.271.937.2251.2411.12601.28325071.p20
13  46+  2.5.17.461.160081.159686609.1445443990517.p21
13  47+  2.7.498851139881.p40
13  48+  2.97.2657.88993.441281.283763713.127028743393.p18
13  49+  2.7^3.29.22079.435709.22896329.54461639.p26
13  50+  2.5^3.17.421.601.641.1253653901.26771688828701.p22
13  51+  2.7.157.617886851384381281.p36
13  52+  2.14281.86113.2176307537.224277684782113.p25
13  53+  2.7.3499.7504072417.202326783229.37379721025854083.p17
13  54+  2.5.17.37.109.28393.428041.1471069.16764949.4220430741361.p19
13  55+  2.7.11^2.2411.53681.128011456717.4122652482568228291.p21

13  56+  2.407865361.17254637799169.p41
13  57+  2.7.157.845083.2657518772948983.6061387217546931661.p21
13  58+  2.5.17.233.20970714732554798304809.p38
13  59+  2.7.5783.p61
13  60+  2.41.4441.8161.14281.29881.217561.815702161.543124566401.p23
13  61+  2.7.6024266671.298681203493.1535617756259.p34
13  62+  2.5.17.1861.1178621.48534593.111109852618983753193.p30
13  63+  2.7^2.19.29.157.271.463.937.22079.11032183.704972647.
         54165939703.p25
13  64+  2.257.3230593.36713826768408543617.p43
13  65+  2.7.11.2411.13417.16381.20333.79301.3808876542352598861.p31
13  66+  2.5.17.5281.28393.37869877.23057835113017.
         3577574298489429481.p24
13  67+  2.7.269.4021.138959.28376556792667.p49
13  68+  2.137.409.14281.63104137.p59
13  69+  2.7.47.157.277.1151.2347.7039.84801400975699.p46
13  70+  2.5^2.17.421.601.641.23161037562937.p54
13  71+  2.7.532615479720542238328159944384931.p46
13  72+  2.1009.3889.680401.407865361.29975087953.659481276875569.
         6654909974864689.p18
13  73+  2.7.45553.64803179963.p65
13  74+  2.5.17.149.1738568407946597.p63
13  75+  2.7.11.31.101.151.157.2411.11551.57751.2113801.28325071.
         966623849742301.3258254426373251.p18
13  76+  2.761.2281.14281.692513.62300665486585624081.p49
13  77+  2.7^2.29.22079.78947177.128011456717.p59
13  78+  2.5.17.313.1873.28393.380329.2874105113569.
         1418792215861230619657.p36
13  79+  2.7.25202746699639.p74
13  80+  2.2657.441281.444641.4335041.283763713.1116130730334721.
         1266394281048641.p29
13  81+  2.7.19.157.163.271.937.30133.73387.904663.1094473.
         259640317.770321341.762615992953.p28
13  82+  2.5.17.1089083758501.1508425553233.p65
13  83+  2.7.167.499.p87
13  84+  2.113.14281.815702161.213867479113.2341071239305009.
         4803378460849459680406337.p27
13  85+  2.7.11.1361.2411.617886851384381281.p69
13  86+  2.5.17.2753.748717.3605871444392247654973.
         14704313671678643430446722121.p35
13  87+  2.7.59.157.1741.45353101.8546789918171.16397414286709.
         388719701504541540919.p35
13  88+  2.4049.407865361.479397861182521358407441.
         1358999792679530608571000081.p35
13  89+  2.7.21001153.2197020880176009487.
         410771620528283009753757952793.p43
13  90+  2.5^2.17.37.421.601.641.28393.428041.1471069.460655521.
         1453046401.14513462249341141.p38
13  91+  2.7^2.29.4733.13417.20333.22079.79301.6176383648709.
         244409090738941856729.p44
13  92+  2.1657.14281.10093085551851597657003882281641.
         13083857523758118007762301611817.p33
13  93+  2.7.157.373.2729.23623.145831193.16389023943543602257.p63
13  94+  2.5.17.36097.75389.99886248944632632917.p74
13  95+  2.7.11.2411.19755615371.3657554614741.5810151294071.
         104422877883960436477.p45
13  96+  2.193.1153.1601.10433.68675120456139881482562689.

         11352931040252580224415980746369.p38
13  97+  2.7.404094629.p99
13  98+  2.5.17.197.2710681.4478993.49517833.139310921.
         4980348180221.23161037562937.p50
13  99+  2.7.19.67.157.271.331.937.1123.6997.122167.960961.
         262119133.626133421.128011456717.p50
13 100+  2.41.7001.14281.29881.128245657201.543124566401.
         38245779934601.538823041855601.52008826491976715201.p27
13 101+  2.7.3327037444864439.7425107270430419.
         9320615531279027221853.14560861044113847497319380951.p30
13 102+  2.5.17^2.1021.28393.897329.1886389.61165661.75094577.
         718862341.6152936617.229090993311472339969.p36
13 103+  2.7.619.89611.1450447.89433871.592045971390131.
         147258804290600290058437.p54
13 104+  2.3329.407865361.161343579388091039418006351540289.p72
13 105+  2.7^2.11.29.31.71.157.463.2411.22079.654221.759641.1759171.
         28325071.54165939703.16566575194331.30152259698778631.p31
13 106+  2.5.17.1061.2034766409871541.55315179851718646236650521.
         1331344930977110852021126518627097.p39
13 107+  2.7.6053633.116077045829.791958624719.p89
13 108+  2.73.433.1297.4177.14281.181297.14799457.815702161.
         9818892432332713.3156176124992605532414981091253561.p34
13 109+  2.7.29867.965087.36081138446572828651824827033.p82
13 110+  2.5^2.17.421.601.641.661.5281.4439770467824561.
         3577574298489429481.202254724516962030101.p51
13 111+  2.7.157.223.21017.152219.1548921490187.10626791079749447.
         3832014914233819776114475402917811351.p44
13 112+  2.2657.441281.283763713.p107
13 113+  2.7.227.9719.1267183.2158975289.4427870101287797057.
         852333015613949705733446242454881.p52
13 114+  2.5.17.229.28393.94621.452660103574249.
         30746232270870161677690909.
         580196961910046805312944783240761.p41
13 115+  2.7.11.47.277.1151.2347.2411.84801400975699.p99
13 116+  2.14281.52201.1142833.9705011473.11779246681.3201194517593.p82
13 117+  2.7.19.157.271.937.13417.20333.79301.5621617.5786353.
         162837793.584288727345658049575114801.p59
13 118+  2.5.17.709.3541.115877.118137589.3172190933.p101
13 119+  2.7^2.29.239.22079.172597045223.2534279978077.
         2810860561573.617886851384381281.p69
13 120+  2.241.1009.407865361.1030309681.2643964801.659481276875569.
         442779263234039928595359287744639041.p51
13 121+  2.7.8713.131891.1801207.2644917463.128011456717.
         236537675815901485856148017522791.p66
13 122+  2.5.17.8626498609074269149060403255784440097670457.p91
13 123+  2.7.83.157.739.638453.247072069.140299545168523469.
         45111380897407574171.915180098788603185080359.p54
13 124+  2.14281.111072257.189616769827415561.304200083494791049.p91
13 125+  2.7.11.101.2411.8501.57751.873251.4831140001.
         3258254426373251.1047326688430383001.p74
13 126+  2.5.17.37.673.2857.4621.15373.28393.428041.1471069.
         23161037562937.61452748127532301.
         1149869468581335216529112593.p49
13 127+  2.7.
         687304059153048444682671733000360314795590589587024964570201596437.
         p75
13 128+  2.96769.2940673.p131

13 129+  2.7.157.173.12566837.7825170961.8001003293.346982008721.
9391016904700369.4048619615131328239.p66
13 130+  2.5^2.17.421.521.601.641.380329.5218721.6691361.
1418792215861230619657.72395677076468070119906108281513841.p56
13 131+  2.7.1049.138599.p137
13 132+  2.89.6073.7129.14281.815702161.134799670920553.
668229430151772307653060812800711311233.p72
13 133+  2.7^2.29.22079.579016463719.104422877883960436477.
7983601174254569126173.p87
13 134+  2.5.17.117628061579381.597891861167008906854329.
208987627610561976973469464852117.p77
13 135+  2.7.11.19.31.157.163.271.937.2251.2411.9181.12601.904663.
28325071.762615992953.4971111387529339441.
19145097390154976011.p58
13 136+  2.407865361.2101969584130840266612144288141743329.
17915834850330581667260806159243454369619165933658481.p55
13 137+  2.7.1714667341.22221138052054464059847263.
252135148303109236253632457534648757143.
315763309485400584429158395865631317867.p40
13 138+  2.5.17.461.28393.160081.159686609.1445443990517.
6533247341521.602053110178724749481.
54836637716450236990971812089.p57
13 139+  2.7.245753.18466388799146717978191.p127
13 140+  2.41.113.281.14281.29881.175362601.960991081.543124566401.
18603406827218262281.4803378460849459680406337.p68
13 141+  2.7.157.498851139881.590202369266263393.
3245178229485124818467952891417691434077.p85
13 142+  2.5.17.569.853.8237.869893.1863893.9598349.
1187807331749399825071347132433.p98
13 143+  2.7.8009.13417.20333.79301.120121.64050562577.128011456717.
26955750986564813756380111933.p86
13 144+  2.97.2017.2657.47521.54721.88993.441281.1590049.283763713.
127028743393.403791981344275297.
8299042833797200969471889569.p61
13 145+  2.7.11.59.1741.2411.7634343961.8546789918171.
16397414286709.943112272332884711.117977510451276076784741.p74
13 146+  2.5.17.293.466462905277.
301557549694923141366115643799231359796226965278135642569.p90
13 147+  2.7^3.29.157.463.22079.435709.22896329.54461639.1314082687.
17712284017.54165939703.14790290915806570339477.
16049801116294358590175881.p53
13 148+  2.3257.14281.12716161.560678027993.p139
13 149+  2.7.2683.2178083.p156
13 150+  2.5^3.17.421.601.641.28393.460655521.1253653901.1453046401.
20858983540201.26771688828701.2152338940237584453701.p76
13 151+  2.7.49529.3457568707.2462954753131.c141
13 152+  2.407865361.p161
13 153+  2.7.19.157.271.307.937.1872876367.8122054267.936052510699.
15840010226107.617886851384381281.
476622264829847630603684799705499201.p61
13 154+  2.5.17.5281.7393.1702933.150324329.23161037562937.
718377597171850001.3577574298489429481.
4209006442599882158485591696242263069.p61
13 155+  2.7.11.373.2411.2729.24181.11268811.145831193.42277560371.
151091386477111.13496217990809681.16389023943543602257.
44630257194378716844161.p59
13 156+  2.14281.86113.815702161.2176307537.224277684782113.

         7009558351001043033313937.
         14269421767320773422797054675027409.p73
13 157+  2.7.1571.9421.1579421.1302884463846205672630741.
         5321704544480702707255714477.42086256219133569812191465703.p80
13 158+  2.5.17.317.p172
13 159+  2.7.157.3499.76003.7504072417.202326783229.1462139329561.
         37379721025854083.39349502420618381.
         197795581824045269616226824820047463.p64
13 160+  2.193.1601.10433.96001.68675120456139881482562689.c138
13 161+  2.7^2.29.47.277.1151.2347.22079.84801400975699.
         105533603618353.c134
13 162+  2.5.17.37.109.28393.428041.1471069.16764949.303474601.
         29320472989.4220430741361.1639740678532467913.p102
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24   3-  23.601
24   5-  23.p6
24   7-  23.29.239.p5
24   9-  19.23.601.2017.4987
24  11-  23.67.7349.p9
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24  19-  23.p25
24  21-  23.29.43.239.601.28771.10426753.p8
24  23-  23^2.47.124799.304751.p17
24  25-  23.101.346201.1661037601.p17
24  27-  19.23.379.601.2017.2377.4987.2400571.p13
24  29-  23.1973.10151.1130131.p26
24  31-  23.311.1613.9859.35279.39619.4045774723.p13
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24  45-  19.23.241.601.2017.4987.17881.24481.346201.p34
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24  51-  23.307.601.1531.2347.120574031.6166060753.341563234253.p28
24  53-  23.p72
24  55-  23.67.7349.346201.434501.134367047.340245936911.
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24  57-  23.457.601.4219.112950414769.7282588256957615350925401.p33
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24  63-  19.23.29.43.239.601.2017.4987.28771.2585521.10426753.
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24  65-  23.53.6553.15913.346201.6895253.1679822951.p58
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24  75-  23.101.241.601.14401.17881.24481.33601.36451.44851.346201.
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24  77-  23.29.67.239.7349.28771.136753.134367047.
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24  79-  23.317.442243.19180727.413846919804499727099665581797563.p60
24  81-  19.23.163.379.601.2017.2377.4987.2400571.3227151869857.
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24  87-  23.523.601.1973.10151.1130131.2054968537.
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24  91-  23.29.53.239.547.6553.15913.28771.81173.6895253.240319171.

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24  93-  23.311.601.1489.1613.9859.35279.39619.137827.4045774723.
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24  95-  23.191.346201.204956205349305191.7282588256957615350925401.
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24  97-  23.4541347.69291447660895432806381824641.
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24  99-  19.23.67.601.2017.4987.7349.3547963.15808321.134367047.
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24 145-  23.1973.3191.10151.346201.1130131.
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24640601.
p95
24 161-  23^2.29.47.239.28771.124799.304751.87557940066983.
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85   1-  2^2.3.7
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191071.409069.538073317.13695512557.584580328399.
406056846216989647742192725321.p86

85   1+  2.43
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24215742276527455609953001.p129

86   1-  5.17
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