New congruences for the Bernoulli numbers. (English) Zbl 0613.10012

If a prime \(p\) divides the numerator of at least one of the Bernoulli numbers \(B_{2k}\) with \(2\leq 2k\leq p-3\), then we say that \(p\) is irregular and the corresponding pairs \((p,2k)\) are irregular pairs. There are several congruences mod \(p\) for \(B_{2k}\) that have been used to find irregular pairs by computer, the most extensive work of this kind (to \(p<125\,000)\) having been done by the second author [Math. Comput. 32, 583–591 (1978; Zbl 0377.10002)]. Now the authors have extended these computations to \(p<150\, 000\) by using some interesting new congruences for \(B_{2k}\). The present paper contains a report on this work and an analysis of the relevant congruences.
The authors have also applied Vandiver’s well-known criterion to show that Fermat’s Last Theorem (FLT) holds for the new irregular primes. Hence FLT is now proved for all exponents up to \(150\,000\).


11B68 Bernoulli and Euler numbers and polynomials
11D41 Higher degree equations; Fermat’s equation
11-04 Software, source code, etc. for problems pertaining to number theory
11R18 Cyclotomic extensions


Zbl 0377.10002
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