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arXiv · 1201.0617

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

Victor J. W. Guo
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Abstract

For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2 \pmod{p^5}, where n is a positive integer and p>3 is a prime.

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Victor J. W. Guo. 2012-07-21. Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers. https://doi.org/10.1080/10652469.2012.715160

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