A refinement of a congruence result by van Hamme and Mortenson
By Zhi-Wei Sun. 2010
47 citations recorded by the source corpus.
Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3\equiv (-1)^{(p-1)/2}p\pmod{p^3}.$$ In this paper we show further that \begin{align*}\sum_{k=0}^{p-1}(4k+1)\binom{-1/2}k^3\equiv &\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3 \\\equiv & (-1)^{(p-1)/2}p+p^3E_{p-3} \pmod{p^4},\end{align*}where $E_0,E_1,E_2,\ldots$ are Euler numbers. We also prove that if $p>3$ then $$\sum_{k=0}^{(p-1)/2}\frac{20k+3}{(-2^{10})^k}\binom{4k}{k,k,k,k}\equiv(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2)\pmod{p^4}.$$
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/745043c00bcee9aa820eb46a97320b0be0d833c8
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
Citations in the corpus
- Some congruences related to a congruence of Van Hamme (2019, background): “[26] Sun Z-W.”
- On Van Hamme's (A.2) and (H.2) supercongruences (2018, methodology): “It is known that Gaussian hypergeometric series (see, for example, [3–6, 12]) and the W-Z method (see, for example, [2, 8, 11, 15]) commonly apply to the Ramanujan-type supercongruences.”