Diophantine equations involving double factorials

TL;DR

This paper explores solutions to Diophantine equations involving double factorials, extending previous factorial-based results, and demonstrates finiteness of solutions under the ABC conjecture for certain polynomial-related equations.

Contribution

It generalizes known factorial equations to double factorials, provides all integer solutions, and establishes finiteness results under the ABC conjecture for polynomial equations involving double factorials.

Findings

All integer solutions for modified factorial equations are identified.

Finiteness of solutions is proven under the ABC conjecture for certain polynomial equations.

Extensions to variations of the original equations are provided.

Abstract

We are motivated by a result of Alzer and Luca who presented all the integer solutions to the relations $(k!)^n-k^n=(n!)^k-n^k$ and $(k!)^n+k^n=(n!)^k+n^k$. We modify the equations by considering the double factorial instead and present all integer solutions. We also consider some variations of these equations. Furthermore, we study equations of the form $f(x)=A_1^{n_1}n_1!!\cdots A_r^{n_r}n_r!!$, where $f(x)$ is a rational polynomial, and show that under the ABC conjecture there are only finitely many integer solutions.