A note on some polynomial-factorial diophantine equations

TL;DR

This paper investigates polynomial-factorial Diophantine equations, exploring conditions under which they have finitely or infinitely many integer solutions, extending classical problems related to factorial equations and polynomial forms.

Contribution

It generalizes previous results by analyzing equations involving factorials, polynomials, and their combinations, including new cases with factorials and homogeneous polynomials.

Findings

Finiteness results under certain polynomial conditions

Extension of classical factorial Diophantine problems

Analysis of equations with factorials, polynomials, and their combinations

Abstract

In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation $n!=x^2-1$. It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed that a weak form of Szpiro's-conjecture implies that Brocard's equation has finitely many integer solutions. More generally, assuming the ABC-conjecture, Luca showed that equations of the form $n!=P(x)$ where $P(x)\in\mathbb{Z}[x]$ of degree $d\geq 2$ have only finitely many integer solutions with $n>0$. And if $P(x)$ is irreducible, Berend and Harmse proved unconditionally that $P(x)=n!$ has only finitely many integer solutions. In this note we study diophantine equations of the form $g(x_1,...,x_r)=P(x)$ where $P(x)\in\mathbb{Z}[x]$ of degree $d\geq 2$ and $g(x_1,...,x_r)\in \mathbb{Z}[x_1,...,x_r]$ where for $x_i$ one may also plug in $A^{n}$ or the Bhargava factorial $n!_S$. We want to understand when there are finitely many or infinitely many integer solutions. Moreover, we study diophantine equations of the form $g(x_1,...,x_r)=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a homogeneous polynomial of degree $\geq2$.