Irregular primes to 163 million

Buhler, J.; Harvey, D.

doi:10.1090/S0025-5718-2011-02461-0

Irregular primes to 163 million
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by J. P. Buhler and D. Harvey PDF

Math. Comp. 80 (2011), 2435-2444

Abstract:

We compute all irregular primes less than 163 577 856. For all of these primes we verify that the Kummer–Vandiver conjecture holds and that the $\lambda$-invariant is equal to the index of irregularity.

References

Joe Buhler, Richard Crandall, Reijo Ernvall, Tauno Metsänkylä, and M. Amin Shokrollahi, Irregular primes and cyclotomic invariants to 12 million, J. Symbolic Comput. 31 (2001), no. 1-2, 89–96. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806208, DOI 10.1006/jsco.1999.1011
J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp. 61 (1993), no. 203, 151–153. MR 1197511, DOI 10.1090/S0025-5718-1993-1197511-5
J. P. Buhler, R. E. Crandall, and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 (1992), no. 200, 717–722. MR 1134717, DOI 10.1090/S0025-5718-1992-1134717-4
A. Bostan, G. Lecerf, and É. Schost, Tellegen’s principle into practice, Symbolic and Algebraic Computation (J. R. Sendra, ed.), ACM Press, 2003, Proceedings of ISSAC’03, Philadelphia, August 2003., pp. 37–44.
L. Bluestein, A linear filtering approach to the computation of discrete Fourier transform, Audio and Electroacoustics, IEEE Transactions on 18 (1970), no. 4, 451–455.
Leo Corry, Fermat meets SWAC: Vandiver, the Lehmers, computers, and number theory, IEEE Ann. Hist. Comput. 30 (2008), no. 1, 38–49. MR 2412202, DOI 10.1109/MAHC.2008.6
2009, http://www.mscs.dal.ca/˜dilcher/bernoulli.html/.
Karl Dilcher, Ladislav Skula, and Ilja Sh. Slavutskiǐ, Bernoulli numbers, Queen’s Papers in Pure and Applied Mathematics, vol. 87, Queen’s University, Kingston, ON, 1991. Bibliography (1713–1990). MR 1119305
R. Ernvall and T. Metsänkylä, Cyclotomic invariants for primes between $125\,000$ and $150\,000$, Math. Comp. 56 (1991), no. 194, 851–858. MR 1068819, DOI 10.1090/S0025-5718-1991-1068819-7
R. Ernvall and T. Metsänkylä, Cyclotomic invariants for primes to one million, Math. Comp. 59 (1992), no. 199, 249–250. MR 1134727, DOI 10.1090/S0025-5718-1992-1134727-7
T. Granlund, The GNU Multiple Precision Arithmetic library, 2008, http://gmplib.org/.
D. Harvey, A multimodular algorithm for computing Bernoulli numbers, Math. Comp. 79 (2010), 2361–2370.
—, The zn_poly library, 2008, http://www.cims.nyu.edu/˜harvey/zn_poly/.
David Harvey, A cache-friendly truncated FFT, Theoret. Comput. Sci. 410 (2009), no. 27-29, 2649–2658. MR 2531107, DOI 10.1016/j.tcs.2009.03.014
David Harvey, Faster polynomial multiplication via multipoint Kronecker substitution, J. Symbolic Comput. 44 (2009), no. 10, 1502–1510. MR 2543433, DOI 10.1016/j.jsc.2009.05.004
—, http://www.cims.nyu.edu/˜harvey/irregular/, 2009.
Guillaume Hanrot, Michel Quercia, and Paul Zimmermann, The middle product algorithm. I, Appl. Algebra Engrg. Comm. Comput. 14 (2004), no. 6, 415–438. MR 2042800, DOI 10.1007/s00200-003-0144-2
Wells Johnson, Irregular primes and cyclotomic invariants, Math. Comp. 29 (1975), 113–120. MR 376606, DOI 10.1090/S0025-5718-1975-0376606-9
Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028, DOI 10.1007/978-1-4612-0987-4
Henri J. Nussbaumer, Fast polynomial transform algorithms for digital convolution, IEEE Trans. Acoust. Speech Signal Process. 28 (1980), no. 2, 205–215. MR 563380, DOI 10.1109/TASSP.1980.1163372
A. Schönhage, Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2, Acta Informat. 7 (1976/77), no. 4, 395–398. MR 0436663, DOI 10.1007/bf00289470
V. Shoup, NTL: A library for doing number theory, http://www.shoup.net/ntl/, 2009.
I. Sh. Slavut⋅skiĭ, A remark on the paper of T. Uehara: “On $p$-adic continuous functions determined by the Euler numbers” [Rep. Fac. Sci. Engrg. Saga Univ. Math. No. 8 (1980), 1–8; MR0567622 (81e:12020)], Rep. Fac. Sci. Engrg. Saga Univ. Math. 15 (1987), no. 1, 1–2. MR 879252
A. Schönhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7 (1971), 281–292 (German, with English summary). MR 292344, DOI 10.1007/bf02242355
Jonathan W. Tanner and Samuel S. Wagstaff Jr., New congruences for the Bernoulli numbers, Math. Comp. 48 (1987), no. 177, 341–350. MR 866120, DOI 10.1090/S0025-5718-1987-0866120-4
H. S. Vandiver, Symmetric functions formed by systems of elements of a finite algebra and their connection with Fermat’s quotient and Bernoulli’s numbers, Ann. of Math. (2) 18 (1917), no. 3, 105–114. MR 1503591, DOI 10.2307/2007115
Joris van der Hoeven, The truncated Fourier transform and applications, ISSAC 2004, ACM, New York, 2004, pp. 290–296. MR 2126956, DOI 10.1145/1005285.1005327
—, Notes on the truncated Fourier transform, unpublished, available from http://www.math.u-psud.fr/˜vdhoeven/, 2005.
Samuel S. Wagstaff Jr., The irregular primes to $125000$, Math. Comp. 32 (1978), no. 142, 583–591. MR 491465, DOI 10.1090/S0025-5718-1978-0491465-4
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7