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arXiv · 1511.05553

Proof of Sun's conjectures on super congruences and the divisibility of certain binomial sums

Guo-Shuai Mao·Tao Zhang
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Abstract

In this paper, we prove two conjectures of Z.-W. Sun: $$2n\binom{2n}n\big|\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}\ \mbox{for}\ \mbox{all}\ n=2,3,\cdots,$$ and $$\sum_{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\binom{2k}{k}^3\equiv p+2\left(\frac{-1}{p}\right)p^3E_{p-3}\pmod{p^4},$$ where $p>3$ is a prime and $E_0,E_1,E_2,\cdots$ are Euler numbers.

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Guo-Shuai Mao, Tao Zhang. 2019-10-29. Proof of Sun's conjectures on super congruences and the divisibility of certain binomial sums. https://doi.org/10.1007/s11139-019-00138-z

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