Diophantine equations involving double factorials

TL;DR

This work extends Alzer–Luca's results by solving Diophantine equations involving double factorials $n!!$, providing complete solutions for several symmetric relations between $(n!!)$ and $(k!!)$. The authors establish key growth properties via Lemma 2.1, perform detailed case analyses to classify all integer solutions for the base equations in Theorems 1.1–1.5, and introduce a general framework $f(x)=A_1^{n_1}n_1!!\cdots A_r^{n_r}n_r!!$ with finiteness results. Moreover, under the ABC conjecture, they prove finiteness for polynomial right-hand sides of degree at least 2, including variants where exponents are replaced by $n_i!$ or $n_i!!$, thereby linking double factorial Diophantine equations to deep conjectures in number theory. The results contribute new exact classifications in the factorial-family of Diophantine problems and demonstrate methods that extend to broader exponential-factorial expressions.

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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark
• Conjecture : ABC-conjecture
• Theorem 1.7
• Remark