On a generalization of the Brocard--Ramanujan Diophantine equation

Saša Novaković

paper

On a generalization of the Brocard--Ramanujan Diophantine equation

arXiv:2602.09687

Abstract

Let $Q_{1}, ..., Q_{r} \in Z [x]$ be polynomials having $0$ as a root. Let $f (x, y) \in Z [x, y]$ be a homogeneous polynomial with factorization $f (x, y) = f_{1} (x, y)^{e_{1}} \dots f_{u} (x, y)^{e_{u}}$ , 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 $\prod_{i = 1}^{r} Q_{i} (A_{i}^{n_{i}} n_{i}!) = f (x, y)$ has finitely many integer solutions.

8 pages, comments are welcome!