Proof of a supercongruence conjectured by Z.-H. Sun

TL;DR

This paper proves a supercongruence conjecture by Z.-H. Sun involving Franel numbers and binomial coefficients, confirming several of Sun's conjectures and extending the understanding of these number theoretic properties.

Contribution

The paper establishes new supercongruences related to Franel numbers, confirming conjectures posed by Z.-H. Sun and providing novel proofs in number theory.

Findings

Confirmed a supercongruence conjecture modulo p^3.

Validated a conjecture on sums involving Franel numbers modulo p^2.

Proved a special case of a conjecture for primes of the form 4k+3.

Abstract

The Franel numbers are defined by $ f_n=\sum_{k=0}^n {n\choose k}^3. $ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv 0\pmod{n{2n\choose n}}, \\ \sum_{k=0}^{p-1}\frac{3k+1}{(-16)^k} {2k\choose k} f_k &\equiv p (-1)^{\frac{p-1}{2}} \pmod{p^3}. \end{align*} where $n>1$ and $p$ is an odd prime. The second congruence modulo $p^2$ confirms a recent conjecture of Z.-H. Sun. We also show that, if $p$ is a prime of the form $4k+3$, then $$ \sum_{k=0}^{p-1}\frac{{2k\choose k} f_k}{(-16)^k} \equiv 0 \pmod p, $$ which confirms a special case of another conjecture of Z.-H. Sun.