A note on balancing binomial coefficients

TL;DR

This paper investigates a specific binomial coefficient balancing equation, proving that the only integer solution occurs at (14,15) using elliptic logarithm methods.

Contribution

It provides a complete solution to a binomial coefficient balancing problem by applying elliptic logarithm techniques to a Diophantine equation.

Findings

The only solution is (14,15).

The method involves elliptic logarithm bounds.

The problem is solved completely for the given equation.

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.