2015/03/11 by Zhongyan Shen, Tianxin Cai, Shen, Zhongyan +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1503.03154
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
In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, ∑_i+j+k=pr\atopi,j,k∈ Pp(1)/(ijk)≡-2pr-1Bp-3 ~(\bmod ~ pr), where Pn denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p,~q and positive integers α,~β, ∑_i+j+k=pαqβ\atopi,j,k\inPpq\atopi≡ j≡ k≡ 1\pmod2(1)/(ijk)≡(7)/(8)(2-q)(1-\frac1q3)pα-1qβ-1Bp-3\pmodpα and ∑_i+j+k=pαqβ\atopi,j,k∈ Ppq\frac(-1)iijk ≡ (1)/(2)(q-2)(1-\frac1q3)pα-1qβ-1Bp-3\pmodpα. Finally, we raise a conjecture that for n>1 and odd prime power pα||n, α≥1, ∑_i+j+k=n\atopi,j,k\inPn\frac(-1)iijk ≡ ∏_q|n\atopq≠ p(1-(2)/(q))(1-\frac1q3)(n)/(2p)Bp-3\pmodpα and ∑_i+j+k=n\atopi,j,k\inPn\atopi≡ j≡ k≡ 1\pmod2(1)/(ijk) ≡ ∏_q|n\atopq≠ p(1-(2)/(q))(1-\frac1q3)(-(7n)/(8p))Bp-3\pmodpα.