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

TL;DR

This paper proves two conjectures by Z.-W. Sun involving super congruences and divisibility properties of specific binomial sums, extending understanding of binomial coefficients and Euler numbers in number theory.

Contribution

The paper establishes the validity of two conjectures on super congruences related to binomial sums and prime modulus, providing new proofs and insights in combinatorial number theory.

Findings

Proved divisibility of a binomial sum by 2n binomial coefficient.

Derived a congruence involving Euler numbers and prime moduli.

Extended Sun's conjectures to new cases and provided rigorous proofs.

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.