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On two congruences involving Apéry and Franel numbers. (English) Zbl 1461.11007

Summary: In this paper, we mainly prove a congruence conjecture of Z.-W. Sun involving Franel numbers: For any prime \(p>3\), we have \[ \sum_{k=0}^{p-1}(-1)^kf_k\equiv \left( \frac{p}{3}\right) +\frac{2p^2}{3}B_{p-2}\left( \frac{1}{3}\right) \pmod{p^3}, \] where \(B_n(x)\) is the \(n\)-th Bernoulli polynomial.

MSC:

11A07 Congruences; primitive roots; residue systems
05A10 Factorials, binomial coefficients, combinatorial functions
11B65 Binomial coefficients; factorials; \(q\)-identities

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References:

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