Irregular primes to one million

Buhler, J. P.; Crandall, R. E.; Sompolski, R. W.

doi:10.1090/S0025-5718-1992-1134717-4

Irregular primes to one million
HTML articles powered by AMS MathViewer

by J. P. Buhler, R. E. Crandall and R. W. Sompolski PDF

Math. Comp. 59 (1992), 717-722 Request permission

Abstract:

Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including their indices of irregularity and associated irregular pairs. Using this data, we verified that Fermat’s "Last Theorem" and Vandiver’s conjecture are true for these primes. Two primes with index of irregularity five were already known; we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six.

References

Mustapha Chellali, Accélération de calcul de nombres de Bernoulli, J. Number Theory 28 (1988), no. 3, 347–362 (French). MR 932380, DOI 10.1016/0022-314X(88)90047-9

The NeXT computer as physics machine

4

Mathematica for the Sciences

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
Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878
K. A. Ribet, On modular representations of $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI 10.1007/BF01231195
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535–541. MR 929551, DOI 10.1090/S0025-5718-1988-0929551-0

The second case of Fermat’s last theorem for fixed irregular prime exponents

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
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
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2