Diophantine equations involving double factorials

2025/10/28 by Novaković, Saša
#11A25 #11A41 #11D99 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.24312

Abstract

We are motivated by a result of Alzer and Luca who presented all the integer solutions to the relations (k!)ⁿ-kⁿ=(n!)^k-n^k and (k!)ⁿ+kⁿ=(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)=A₁^n₁n₁!!⋯ A_r^n_rn_r!!, where f(x) is a rational polynomial, and show that under the ABC conjecture there are only finitely many integer solutions.

Citations

• Some Diophantine equations involving arithmetic functions and Bhargava factorials
• A note on some polynomial-factorial diophantine equations

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