Some Diophantine equations involving arithmetic functions and Bhargava factorials

TL;DR

This paper extends previous results on the finiteness of solutions to certain factorial-based Diophantine equations, including cases with multiple factorials, various arithmetic functions, and Bhargava factorials.

Contribution

It generalizes earlier finiteness results to equations with multiple factorials, new divisor functions, and Bhargava factorials, broadening the scope of known solutions.

Findings

Finitely many solutions for equations with multiple factorials and arithmetic functions.

Inclusion of sum of powers of divisors function in the finiteness results.

Extension of results to equations involving Bhargava factorials.

Abstract

F. Luca proved for any fixed rational number $\alpha>0$ that the Diophantine equations of the form $\alpha\,m!=f(n!)$, where $f$ is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions $(m,n)$. In this paper we generalize the mentioned result and show that Diophantine equations of the form $\alpha\,m_1!\cdots m_r!=f(n!)$ have finitely many integer solutions, too. In addition, we do so by including the case $f$ is the sum of $k$\textsuperscript{th} powers of divisors function. Moreover, we observe that the same holds by replacing some of the factorials with certain examples of Bhargava factorials.