Illinois Journal of Mathematics

A refinement of a congruence result by van Hamme and Mortenson

Zhi-Wei Sun

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Let $p$ be an odd prime. In 2008, E. Mortenson proved van Hamme’s following conjecture:

\[\sum_{k=0}^{(p-1)/2}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}\equiv(-1)^{(p-1)/2}p\ \bigl(\operatorname{mod}p^{3}\bigr).\]

In this paper, we show further that

\begin{eqnarray*}\sum_{k=0}^{p-1}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}&\equiv&\sum_{k=0}^{(p-1)/2}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}\\[-2pt]&\equiv&(-1)^{(p-1)/2}p+p^{3}E_{p-3}\ \bigl(\operatorname{mod}p^{4}\bigr),\end{eqnarray*}

where $E_{0},E_{1},E_{2},\ldots$ are Euler numbers. We also prove that if $p>3$ then

\begin{eqnarray*}&&\sum_{k=0}^{(p-1)/2}\frac{20k+3}{(-2^{10})^{k}}\Bigl({\matrix{4k\\k,k,k,k}}\Bigr)\\&&\quad \equiv(-1)^{(p-1)/2}p\bigl(2^{p-1}+2-\bigl(2^{p-1}-1\bigr)^{2}\bigr)\ \bigl(\operatorname{mod}p^{4}\bigr).\end{eqnarray*}

Article information

Illinois J. Math., Volume 56, Number 3 (2012), 967-979.

First available in Project Euclid: 31 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx] 11A07: Congruences; primitive roots; residue systems 11B68: Bernoulli and Euler numbers and polynomials


Sun, Zhi-Wei. A refinement of a congruence result by van Hamme and Mortenson. Illinois J. Math. 56 (2012), no. 3, 967--979. doi:10.1215/ijm/1391178558.

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