On two congruence conjectures of Z.-W. Sun involving Franel numbers

Abstract

In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb{Z}$ and $x\equiv 1(\mathrm{mod}3)$, then $$x\equiv \frac{1}{4}\sum _{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv \frac{1}{2}\sum _{k=0}^{p-1}(3k+2)\frac{f_k}{(-4{)}^k}(\mathrm{mod}{p}^2),$$ and if $p\equiv 1(\mathrm{mod}3)$, then $$\sum _{k=0}^{p-1}\frac{f_k}{2^k}\equiv \sum _{k=0}^{p-1}\frac{f_k}{(-4{)}^k}(\mathrm{mod}{p}^3),$$ where $f_n={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^3$ stands for the $n$th Franel number.

20 pages, revised some typos