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Irregular primes and cyclotomic invariants to 12 million. (English) Zbl 1001.11061

Let $p$ be an odd prime. A pair $\left(p,2t\right)$ $\left(1\le t\le \left(p-3\right)/2$, $t\in ℤ$) is said to be irregular for $p$ if $p$ divides the Bernoulli number ${B}_{2t}$. The number $i\left(p\right)$ of irregular pairs of $p$ is called the index of irregularity of $p$. The prime $p$ is regular in case $i\left(p\right)=0$ and if $i\left(p\right)\ge 1$, $p$ is irregular.

In 1857 E. E. Kummer had found out that the primes 37, 59, and 67 are irregular, and in 1879 he made the computation of irregular primes up to 163 (probably by hand). Since then, many mathematicians have continued these computations using better computational tools (calculators, computers) using increasingly better and more effective methods.

The presented results on computations of $i\left(p\right)$ for $p$ up to 12 million use two different algorithms. The first one is based on the power series method combined with enhanced multisectioning and convolution algorithms used in the last tables by the first four authors [Math. Comput. 61, 151-153 (1993; Zbl 0789.11020)]. The second method is a novel approach originated in the study of Stickelberger codes in [M. A. Shokrollahi, Des. Codes Cryptography 9, 203-213 (1996; Zbl 0866.94022)].

In this paper the indices of irregularity are given for primes up to 12 million. The index $i\left(p\right)$ for these primes equal 0 to 7. Three new irregular primes with this index equal to 7 were found to one known prime with this property. Further, the Kummer-Vandiver conjecture was verified, that is the class number of the field $ℚ\left(cos\left(2\pi /p\right)\right)$ is prime to $p$. No counterexample was found. At the conclusion the cyclotomic invariants were calculated.

##### MSC:
 11Y40 Algebraic number theory computations 11-04 Machine computation, programs (number theory)