Wolstenholme and Vandiver primes
| Autor: | Tim Trudgian, Shehzad Hathi, Andrew R. Booker, Michael J. Mossinghoff |
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| Rok vydání: | 2021 |
| Předmět: |
Algebra and Number Theory
Mathematics - Number Theory Mathematics::Number Theory Congruence relation Prime (order theory) Combinatorics Bernoulli's principle symbols.namesake Number theory Wolstenholme prime FOS: Mathematics symbols 11B68 11Y40 11A41 Number Theory (math.NT) Euler number Bernoulli number Mathematics |
| Zdroj: | The Ramanujan Journal. 58:913-941 |
| ISSN: | 1572-9303 1382-4090 |
| DOI: | 10.1007/s11139-021-00438-3 |
| Popis: | 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. 26 pages; to appear in Ramanujan J |
| Databáze: | OpenAIRE |
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