# On the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$

## TL;DR

This paper completely solves a class of Diophantine equations involving binomial coefficients and a small integer shift by analyzing integral points on elliptic and hyperelliptic curves, providing new computational and theoretical insights.

## Contribution

It introduces a method to solve the binomial coefficient Diophantine equation for specific parameters by finding all integral points on related algebraic curves, expanding understanding of these equations.

## Key findings

- Complete solutions for the specified binomial equations with -3 ≤ d ≤ 3.
- Identification of all integral points on certain elliptic and hyperelliptic curves.
- Additional computational and theoretical observations related to the equations.

## Abstract

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\; (3,6),\; (4,6), \; (4,8)\}.$ Moreover, we present some other observations of computational and theoretical nature concerning the title equation.