Powers of two weighted sum of the first p divided Bernoulli numbers modulo p

2020/01/09 by Claire Levaillant, Levaillant, Claire · 1 citation
Mathematics · #05A05 #05A19 #11Y11 #11Y40 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #Number Theory (math.NT) #acm:05A05 #acm:05A19 #acm:11Y11 #acm:11Y40 #math.CO #math.NT #msc:05A05 #msc:05A19 #msc:11Y11 #msc:11Y40

paper · pdf · doi:10.48550/arxiv.2001.03471

26 pages, 24 references; new version: typo in equation numbering corrected

openalex publication_date 2020/01/09 · arxiv created 2021/11/05 · arxiv updated 2021/11/08 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p² divides the Fermat quotient q_p(2).

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