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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 22kp-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<125000) 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<150000 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 150000.


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