*(English)*Zbl 0613.10012

If a prime $p$ divides the numerator of at least one of the Bernoulli numbers ${B}_{2k}$ 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 ${B}_{2k}$ that have been used to find irregular pairs by computer, the most extensive work of this kind (to $p<125\phantom{\rule{0.166667em}{0ex}}000)$ having been done by the second author [Math. Comput. 32, 583–591 (1978; Zbl 0377.10002)]. Now the authors have extended these computations to $p<150\phantom{\rule{0.166667em}{0ex}}000$ 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 $150\phantom{\rule{0.166667em}{0ex}}000$.

##### MSC:

11B68 | Bernoulli and Euler numbers and polynomials |

11D41 | Higher degree diophantine equations |

11-04 | Machine computation, programs (number theory) |

11R18 | Cyclotomic extensions |