A New Super Congruence Involving Multiple Harmonic Sums

2014/10/07 by Liuquan Wang, Wang, Liuquan
Mathematics · #11A41 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Primary 11A07 #math.NT #msc:11A07 #msc:11A41

paper · pdf · doi:10.48550/arxiv.1410.1712

10 pages

openalex publication_date 2014/10/07 · arxiv created 2014/10/12 · arxiv updated 2014/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P_n denote the set of positive integers which are prime to n. Let B_n be the n-th Bernoulli number. For any prime p≥ 5 and r≥ 2, we prove that ∑_\beginsmallmatrix l₁+l₂+⋯ +l₅=p^r l₁,⋯ ,l₅∈ P_p \endsmallmatrix\frac1l₁l₂l₃l₄l₅≡ -(5!)/(6)B_p-5p^r-1 \pmodp^r. This gives an extension of a family of super congruences found by Wang, Cai and Zhao.

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