2021/01/27 by Andrew R. Booker, Booker, Andrew R., Shehzad Hathi +5
Mathematics · #11A41 #11B68 #11Y40 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2101.11157
openalex publication_date 2021/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A prime p is a Wolstenholme prime if \binom2pp≡2 mod p4, or, equivalently, if p divides the numerator of the Bernoulli number Bp-3; a Vandiver prime p is one that divides the Euler number Ep-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 1011, 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.