A refinement of the Hamme-Mortenson congruence

Details A refinement of the Hamme-Mortenson congruence A refinement of the Hamme-Mortenson congruence

2010-11-08

arxiv.org/abs/1011.1902v5

Mathematics

Number Theory

13 pages. Polished version

Scientific paper

Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}^{(p-1)/2}(4k+1)\binom(-1/2,k)^3=(-1)^{(p-1)/2}p (mod p^3).$$ In this paper we show further that $$\sum_{k=0}^{p-1}(4k+1)\binom(-1/2,k)^3 =\sum_{k=0}^{(p-1)/2}(4k+1)\binom(-1/2,k)^3=(-1)^{(p-1)/2}p+p^3E_{p-3} (mod p^4),$$ where $E_0,E_1,E_2,...$ are Euler numbers. We also prove that if $p>3$ then $$\sum_{k=0}^{(p-1)/2}(20k+3)/(-2^{10})^k*\binom{4k}{k,k,k,k} =(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2) (mod p^4).$$

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