Repeated Binomial Coefficients and High-Degree Curves

TL;DR

This paper investigates solutions to binomial coefficient equations involving shifts, proving finiteness of solutions in certain cases and exploring their potential role in addressing Singmaster's conjecture on repeated binomial coefficients.

Contribution

It introduces new results on the solutions of shifted binomial coefficient equations and applies Diophantine geometry to prove finiteness in specific cases, offering a novel approach to Singmaster's conjecture.

Findings

Solutions are finite when $a eq b$

A ratio approximation for solutions is established

Potential utility in proving Singmaster's conjecture

Abstract

We consider the problem of characterizing solutions in $(x, y)$ to the equation ${x \choose y}={{x-a} \choose {y+b}}$ in terms of $a$ and $b$. We obtain one simple result which allows the determination of a ratio in terms of $a$ and $b$ which the ratio $\frac{x}{y}$ must approximate. We then add to the understanding of the infinite family of repeated coefficients discovered by D. Singmaster, by using fundamental results from Diophantine geometry to prove that in the case $a \neq b$, solutions to ${x \choose y}={{x-a} \choose {y+b}}$ are finite. Finally, we make some observations about the potential utility of equations of the form ${x \choose y}={{x-a} \choose {y+b}}$ in proving Singmaster's conjecture, which is the main unsolved problem in the area of repeated binomial coefficient study. We remark that this approach to the conjecture is markedly different from previous approaches, which have only established logarithmic bounds on a function which counts the number of representations of $t$ as a binomial coefficient.