Buhler, J.; Crandall, R.; Ernvall, Reijo; Metsänkylä, Tauno Irregular primes and cyclotomic invariants to four million. (English) Zbl 0789.11020 Math. Comput. 61, No. 203, 151-153 (1993). The first two authors and R. W. Sompolski [Math. Comput. 59, 717- 722 (1992; Zbl 0768.11009)] previously calculated all irregular pairs of primes less than 1 million. The present authors extend this to 4 million. They verify that, up to this bound, the following are true: Fermat’s Last Theorem, Vandiver’s Conjecture, and the Iwasawa \(\lambda\)-invariant equals the index of irregularity. One example of a prime with index of irregularity 7 was found, the previous highest being 6. Reviewer: L.Washington (College Park) Cited in 4 ReviewsCited in 22 Documents MSC: 11D41 Higher degree equations; Fermat’s equation 11R23 Iwasawa theory 11-04 Software, source code, etc. for problems pertaining to number theory 11B68 Bernoulli and Euler numbers and polynomials 65Y20 Complexity and performance of numerical algorithms 11Y55 Calculation of integer sequences 68Q25 Analysis of algorithms and problem complexity Keywords:irregular primes; Fermat’s last theorem; Vandiver’s conjecture; Iwasawa \(\lambda\)-invariant; index of irregularity Citations:Zbl 0768.11009 PDFBibTeX XMLCite \textit{J. Buhler} et al., Math. Comput. 61, No. 203, 151--153 (1993; Zbl 0789.11020) Full Text: DOI Online Encyclopedia of Integer Sequences: Smallest prime of irregularity index n. Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4). References: [1] J. P. Buhler, R. E. Crandall, and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 (1992), no. 200, 717 – 722. · Zbl 0768.11009 [2] D. Bernstein, Multidigit multiplication, the FFT, and Nussbaumer’s algorithm, manuscript. [3] Richard Crandall and Barry Fagin, Discrete weighted transforms and large-integer arithmetic, Math. Comp. 62 (1994), no. 205, 305 – 324. · Zbl 0839.11065 [4] R. Ernvall and T. Metsänkylä, Cyclotomic invariants for primes to one million, Math. Comp. 59 (1992), no. 199, 249 – 250. · Zbl 0760.11029 [5] Wells Johnson, Irregular primes and cyclotomic invariants, Math. Comp. 29 (1975), 113 – 120. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. · Zbl 0302.10020 [6] Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. · Zbl 0477.65002 [7] D. H. Lehmer, Emma Lehmer, and H. S. Vandiver, An application of high-speed computing to Fermat’s last theorem, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 25 – 33. · Zbl 0055.04004 [8] Jonathan W. Tanner and Samuel S. Wagstaff Jr., New congruences for the Bernoulli numbers, Math. Comp. 48 (1987), no. 177, 341 – 350. · Zbl 0613.10012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.