A note on balancing binomial coefficients

Abstract

In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation $$\binom{1}{5}+\binom{2}{5}+...+\binom{x-1}{5}=\binom{x+1}{5}+...+\binom{y}{5}$$ where $y>x>5$ are integer unknowns. We prove that the only integral solution is $(x,y)=(14,15)$. Our method is mainly based on the linear form in elliptic logarithms.