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arXiv · 2101.11157

Wolstenholme and Vandiver primes

Andrew R. Booker·Shehzad Hathi·Michael J. Mossinghoff·Timothy S. Trudgian
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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.

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Andrew R. Booker, Shehzad Hathi, Michael J. Mossinghoff, Timothy S. Trudgian. 2021-01-27. Wolstenholme and Vandiver primes. https://arxiv.org/abs/2101.11157

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