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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.6 P.946-948


Congruences for finite triple harmonic sums

Author(s):  FU Xu-dan, ZHOU Xia, CAI Tian-xin

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310028, China; more

Corresponding email(s):   fuxudan@126.com

Key Words:  Finite triple harmonic sums, Recursive relation, Bernoulli numbers, Catalan numbers

FU Xu-dan, ZHOU Xia, CAI Tian-xin. Congruences for finite triple harmonic sums[J]. Journal of Zhejiang University Science A, 2007, 8(6): 946-948.

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publisher="Zhejiang University Press & Springer",

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T1 - Congruences for finite triple harmonic sums
A1 - FU Xu-dan
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DOI - 10.1631/jzus.2007.A0946

Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡−2Bp−3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β,γ;p) (mod p) is considered for all positive integers α,β,γ. We refer to w=α+β+γ as the weight of the sum, and show that if w is even, S(α,β,γ;p)≡0 (mod p) for pw+3; if w is odd, S(α,β,γ;p)≡rBpw (mod p) for pw, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1] Graham, R.L., Knuth, D.E., Patashnik, O., 1994. Concrete Mathematics (2nd Ed.). Addison-Wesley.

[2] Hoffman, M.E., 2004. Quasi-symmetric Functions and Mod p Multiple Harmonic Sums. Http://arxiv.org/abs/math.NT/0401319

[3] Ji, C.G., 2005. A simple proof of a curious congruence by Zhao. Proc. Amer. Math. Soc., 133:3469-3472.

[4] Zhao, J.Q., 2003a. Bernoulli Numbers, Wolstenholme’s Theorem, and p5 Variations of Lucas’ Theorem. Http://arxiv.org/abs/math.NT/0303332, V1.

[5] Zhao, J.Q., 2003b. Partial Sums of Multiple Zeta Value Series I: Generalizations of Wolstenholme’s Theorem. Http://arxiv.org/abs/math.NT/0301252, V2.

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