On two supercongruences of truncated hypergeometric series $${}_{4}F_{3}$$

In this paper, we prove two supercongruences conjectured by Sun via the Wilf–Zeilberger method. One of them is, for any prime $$p>3$$ , $$\begin{aligned} {}_{4}F_{3}\bigg [\begin{array}{llll} \frac{7}{6}&{}\frac{1}{2}&{}\frac{1}{2}&{}\frac{1}{2}\\ &{}\frac{1}{6}&{}1&{}1\end{array}\bigg |\frac{1}{4}\bigg ]_{p-1}\equiv p(-1)^{(p-1)/2}-p^{3}E_{p-3}\pmod {p^{4}}, \end{aligned}$$ where $$E_{p-3}$$ is the $$(p-3)$$ th Euler number. In fact, this supercongruence is a generalization of a supercongruence of van Hamme.

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