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Guo","submitted_at":"2019-03-09T08:57:30Z","abstract_excerpt":"We establish some supercongruences related to a supercongruence of Van Hamme, such as \\begin{align*} \\sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\\frac{(-\\frac{1}{2})_k^3}{k!^3} &\\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\\pmod{p^{4}},\\\\ \\sum_{k=0}^{(p+1)/2} (4k-1)^5 \\frac{(-\\frac{1}{2})_k^4}{k!^4} &\\equiv 16p\\pmod{p^{4}}, \\end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\\tt Sigma} developed by Schneider. 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