Proof of Sun's conjecture on the divisibility of certain binomial sums

TL;DR

This paper proves a conjecture by Z.-W. Sun that a specific binomial sum is divisible by a particular binomial coefficient expression, confirming a long-standing mathematical hypothesis.

Contribution

The paper provides a rigorous proof of Sun's conjecture on the divisibility of a binomial sum, establishing a new result in combinatorial number theory.

Findings

Confirmed Sun's conjecture on divisibility

Showed the sum is divisible by the binomial coefficient expression

Enhanced understanding of binomial sum divisibility properties

Abstract

In this paper, we prove the following result conjectured by Z.-W. Sun: $$ (2n-1){3n\choose n}| \sum_{k=0}^{n}{6k\choose 3k}{3k\choose k}{6(n-k)\choose 3(n-k)}{3(n-k)\choose n-k}. $$ by showing that the left-hand side divides each summand on the right-hand side.