Wolstenholme and Vandiver primes

A prime p is a Wolstenholme prime if 2 p p ≡ 2 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...

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Bibliographic Details
Published in:The Ramanujan journal 2022, Vol.58 (3), p.913-941
Main Authors: Booker, Andrew R., Hathi, Shehzad, Mossinghoff, Michael J., Trudgian, Timothy S.
Format: Article
Language:English
Subjects:
Combinatorics
Congruences
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
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Summary:A prime p is a Wolstenholme prime if 2 p p ≡ 2 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 ten, and show that no additional Wolstenholme primes exist up to 10 11 , and in the second case by a factor of twenty, 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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-021-00438-3