Congruences for Franel numbers

TL;DR

This paper explores new congruences involving Franel numbers, establishing several prime modulus relations that deepen understanding of their number-theoretic properties.

Contribution

It systematically derives fundamental congruences for Franel numbers modulo prime powers, a novel contribution to their number-theoretic analysis.

Findings

Established congruences for sums involving Franel numbers modulo p^2.

Proved specific sum identities involving alternating signs and reciprocals of k.

Extended the understanding of Franel numbers in modular arithmetic contexts.

Abstract

The Franel numbers given by $f_n=\sum_{k=0}^n\binom{n}{k}^3$ ($n=0,1,2,\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime $p>3$ the following congruences: \begin{align*}\sum_{k=0}^{p-1}(-1)^kf_k&\equiv\left(\frac p3\right)\ \ (\mbox{mod}\ p^2), \\ \sum_{k=0}^{p-1}(-1)^k\,kf_k&\equiv-\frac 23\left(\frac p3\right)\ \ (\mbox{mod}\ p^2), \\ \sum_{k=1}^{p-1}\frac{(-1)^k}kf_k &\equiv0\ \ (\mbox{mod}\ p^2), \\ \sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}f_k&\equiv0\ \ (\mbox{mod}\ p). \end{align*}