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

TL;DR

This paper proves two conjectures by Z.-W. Sun relating prime representations and congruences involving Franel numbers, extending known results with new modular identities and conditions.

Contribution

It provides the first proofs of Sun's conjectures connecting prime representations and Franel number congruences, introducing novel modular identities.

Findings

Established congruences for primes of the form p=x^2+3y^2 with x≡1 mod 3.

Proved congruences involving sums of Franel numbers modulo p^2 and p^3.

Extended the understanding of number-theoretic properties of 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 $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{(-4)^k}\pmod{p^2},$$ and if $p\equiv1\pmod3$, then $$\sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{(-4)^k}\pmod{p^3},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ stands for the $n$th Franel number.