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

Research paper by **Victor J. W. Guo**

Indexed on: **28 Apr '14**Published on: **28 Apr '14**Published in: **Mathematics - Number Theory**

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#### 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.