On a generalization of the Brocard--Ramanujan Diophantine equation

TL;DR

This paper studies a generalized form of the Brocard--Ramanujan Diophantine equation involving polynomial products and homogeneous polynomials, proving finiteness of solutions under certain conditions.

Contribution

It extends the classical Brocard--Ramanujan equation to a broader class involving polynomial products and homogeneous polynomials, establishing finiteness results.

Findings

Finitely many solutions exist under specified conditions.

Generalization encompasses broader polynomial and factorial structures.

Provides new insights into polynomial Diophantine equations.

Abstract

Let $Q_1,...,Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in\mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y)^{e_1}\cdots f_u(x,y)^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous polynomials of degree $d_i\geq 2$. Fix some positive integers $A_1,...,A_r$. We show that under certain conditions, the diophantine equation $\prod_{i=1}^rQ_i(A_i^{n_i}n_i!)=f(x,y)$ has finitely many integer solutions.