Super congruences involving alternating harmonic sums modulo prime powers

2015/03/11 by Zhongyan Shen, Tianxin Cai, Shen, Zhongyan +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1503.03156

arxiv created 2015/03/11 · openalex publication_date 2015/03/11 · arxiv updated 2015/03/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, ∑_i+j+k=p^r\atopi,j,k∈ P_p(1)/(ijk)≡-2p^r-1B_p-3 (\bmod p^r), where P_n denote the set of positive integers which are prime to n. In this note, we establish a combinational congruence of alternating harmonic sums for any odd prime p and positive integers r, ∑_i+j+k=p^r\atopi,j,k∈ P_p\frac(-1)ⁱijk ≡ (1)/(2)p^r-1B_p-3 (\bmod p^r). For any odd prime p≥ 5 and positive integers r, we have 4∑_i₁+i₂+i₃+i₄=2p^r\atopi₁, i₂, i₃, i₄∈ P_p\frac(-1)^i₁i₁i₂i₃i₄+3∑_i₁+i₂+i₃+i₄=2p^r\atopi₁, i₂, i₃, i₄∈ P_p\frac(-1)^i₁+i₂i₁i₂i₃i₄
≡\begincases (216)/(5)pB_p-5\pmodp², if r=1,
(36)/(5)p^rB_p-5\pmodp^r+1, if r>1.
\endcasesFor any odd prime p> 5 and positive integers r, we have ∑_i₁+i₂+i₃+i₄+i₅=2p^r\atopi₁, i₂, i₃, i₄, i₅∈ P_p\frac(-1)^i₁i₁i₂i₃i₄i₅+2∑_i₁+i₂+i₃+i₄+i₅=2p^r\atopi₁, i₂, i₃, i₄, i₅∈ P_p\frac(-1)^i₁+i₂i₁i₂i₃i₄i₅
≡\begincases 12B_p-5\pmodp, if r=1,
6p^r-1B_p-5\pmodp^r, if r>1. \endcases

Citations

• A Curious Congruence Involving Alternating Harmonic Sums

Related

• A congruence involving alternating harmonic sums modulo p^αq^β
• A Curious Congruence Involving Alternating Harmonic Sums
• On a congruence involving harmonic series and Bernoulli numbers
• Some new curious congruences involving multiple harmonic sums
• A New Super Congruence Involving Multiple Harmonic Sums