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

TL;DR

This paper proves two conjectures by Z.-W. Sun concerning specific congruences involving Franel numbers, revealing new properties and congruence relations for these combinatorial sequences.

Contribution

The paper confirms two conjectures on congruences for Franel numbers, providing new insights into their arithmetic properties and extending known results.

Findings

Proved that ^{n-1}(3k+2)(-1)^k f_k \,\equiv\ 0 \pmod{2n^2}

Established that for prime p>3, ^{p-1}(3k+2)(-1)^k f_k \,\equiv\ 2p^2 (2^p-1)^2 \pmod{p^5}

Enhanced understanding of congruence relations for Franel numbers.

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.