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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 \(150\,000\).

MSC:

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

Citations:

Zbl 0377.10002
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