Diophantine equations involving arithmetic functions and factorials

Abstract: F. Luca proved for any fixed rational number $\alpha>0$ that the Diophantine equations $\alpha\,m!=f(n!)$, where $f$ is either the Euler function, the divisor sum function, or the function counting the number of divisors, have finitely many integer solutions in~$m$ and~$n$. In joint work with Novakovi\'{c} 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, the same holds by replacing some of the factorials with certain examples of Bhargava factorials.

Mathematics

Audience: researchers in the topic

( paper )

Combinatorial and additive number theory (CANT 2025)