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

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 $$\sum _{k=0}^{(p-1)/2}\frac{3k+1}{16^k}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3\equiv p+2\left(\frac{-1}{p}\right)p^3{E}_{p-3}(\mathrm{mod}{p}^4),$$ where $p>3$ is a prime and $E_0,E_1,E_2,\cdots$ are Euler numbers.

11 pages

Cited by in corpus (3)

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