Abstract: Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡−2B_p−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 p≥w+3; if w is odd, S(α,β,γ;p)≡rB_p−w (mod p) for p≥w, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.

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

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