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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 $B\sb{2k}$ with $2\le 2k\le p-3$, then we say that $p$ is irregular and the corresponding pairs $(p,2k)$ are irregular pairs. There are several congruences mod $p$ for $B\sb{2k}$ 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 $B\sb{2k}$. 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 $150\,000$.

11B68Bernoulli and Euler numbers and polynomials
11D41Higher degree diophantine equations
11-04Machine computation, programs (number theory)
11R18Cyclotomic extensions
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