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A refinement of a congruence result by van Hamme and Mortenson (1011.1902v6)

Published 8 Nov 2010 in math.NT and math.CO

Abstract: Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}{(p-1)/2}(4k+1)\binom{-1/2}k3\equiv (-1){(p-1)/2}p\pmod{p3}.$$ In this paper we show further that \begin{align*}\sum_{k=0}{p-1}(4k+1)\binom{-1/2}k3\equiv &\sum_{k=0}{(p-1)/2}(4k+1)\binom{-1/2}k3 \\equiv & (-1){(p-1)/2}p+p3E_{p-3} \pmod{p4},\end{align*}where $E_0,E_1,E_2,\ldots$ are Euler numbers. We also prove that if $p>3$ then $$\sum_{k=0}{(p-1)/2}\frac{20k+3}{(-2{10})k}\binom{4k}{k,k,k,k}\equiv(-1){(p-1)/2}p(2{p-1}+2-(2{p-1}-1)2)\pmod{p4}.$$

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