Proof of two supercongruences of truncated hypergeometric series \({}_4 F_3\). (English) Zbl 1539.11010
Summary: In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime \(p > 3\),
\[
{}_4F_3
\left[
\begin{matrix}
\frac{7}{6} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
& \frac{1}{6} & 1 & 1
\end{matrix} \Bigg| {-\frac{1}{8}} \right]_{\frac{p - 1}{2}} \equiv p \left(\frac{-2}{p}\right) + \frac{p^3}{4} \left(\frac{2}{p} \right) E_{p - 3} \quad \pmod{p^4},
\]
where \((\frac{\cdot}{p})\) stands for the Legendre symbol, and \(E_n\) is the \(n\)-th Euler number.
MSC:
| 11A07 | Congruences; primitive roots; residue systems |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
Keywords:
supercongruence; truncated hypergeometric series; Wilf-Zeilberger method; Euler numbers; Legendre symbolSoftware:
SIGMAReferences:
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