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 $$\left(\genfrac{}{}{0ex}{}{1}{5}\right)+\left(\genfrac{}{}{0ex}{}{2}{5}\right)+...+\left(\genfrac{}{}{0ex}{}{x-1}{5}\right)=\left(\genfrac{}{}{0ex}{}{x+1}{5}\right)+...+\left(\genfrac{}{}{0ex}{}{y}{5}\right)$$ 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.

Several minor corrections. Final version published in Proc. Japan Acad. Ser. A Math. Sci