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:

$$\begin{aligned} 2n\left( {\begin{array}{c}2n\\ n\end{array}}\right) \bigg |\sum _{k=0}^{n-1}(3k+1)\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^3{16}^{n-1-k}\quad \text{ for } \text{ all }\ n=2,3,\ldots , \end{aligned}$$

and

$$\begin{aligned} \sum _{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^3\equiv p+2\left( \frac{-1}{p}\right) p^3E_{p-3}\ ({\mathrm{mod}}\ p^4), \end{aligned}$$

where $p>3$ is a prime and $E_0,E_1,E_2,\ldots$ are Euler numbers.