Congruences for Franel numbers
arXiv:1112.1034
Abstract
The Franel numbers given by () 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 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*}
12 pages. Final published version