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Zbl 0613.10012
Tanner, Jonathan W.; Wagstaff, Samuel S.jun
New congruences for the Bernoulli numbers. (English)
[J] Math. Comput. 48, 341-350 (1987). ISSN 0025-5718; ISSN 1088-6842/e

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. \par 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 $150\,000$.
[Tauno Metsänkylä (Turku)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11D41 Higher degree diophantine equations
11-04 Machine computation, programs (number theory)
11R18 Cyclotomic extensions

Keywords: computational number theory; Vandiver's congruence; Bernoulli numbers; Fermat's Last Theorem; irregular primes

Citations: Zbl 0377.10002

Cited in: Zbl 0768.11009 Zbl 0724.11052

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