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.

中文翻译：

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截断的超几何级数$$ {} _ {4} F_ {3} $$ 4 F 3的两个超同余

在本文中，我们通过Wilf-Zeilberger方法证明了Sun推测的两个超同余。其中之一是，对于任何素数\（p> 3 \），

$$ \ begin {aligned} {} _ {4} F_ {3} \ bigg [\ begin {array} {llll} \ frac {7} {6}＆{} \ frac {1} {2}＆{} \ frac {1} {2}＆{} \ frac {1} {2} \\＆{} \ frac {1} {6}＆{} 1＆{} 1 \ end {array} \ bigg | \ frac { 1} {4} \ bigg] _ {p-1} \ equiv p（-1）^ {（p-1）/ 2} -p ^ {3} E_ {p-3} \ pmod {p ^ {4 }}，\ end {aligned} $$

其中\（E_ {p-3} \）是第（（p-3）\）个欧拉数。实际上，这种超一致性是van Hamme的超一致性的推广。