Wolstenholme and Vandiver primes

TL;DR

This paper extends the computational search for Wolstenholme and Vandiver primes, confirming no new primes of these types exist up to 10^11 and 10^11 respectively, using new congruences and GPU-based searches.

Contribution

The authors develop new congruences for Bernoulli and Euler numbers and implement parallel GPU algorithms to efficiently search for these primes.

Findings

No additional Wolstenholme primes up to 10^11.

No additional Vandiver primes up to 10^11.

Enhanced computational methods for prime searches.

Abstract

A prime $p$ is a Wolstenholme prime if $\binom{2p}{p}\equiv2$ mod $p^4$, or, equivalently, if $p$ divides the numerator of the Bernoulli number $B_{p-3}$; a Vandiver prime $p$ is one that divides the Euler number $E_{p-3}$. Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of $10$, and show that no additional Wolstenholme primes exist up to $10^{11}$, and in the second case by a factor of $20$, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruences for Bernoulli and Euler numbers mod $p$ that are favorable for computation, and we implement some highly parallel searches using GPUs.