The Diophantine equation $P(x)=\overset{r}{\underset{i=1}{\prod}}H_{n_i}$

TL;DR

This paper investigates the solutions of polynomial equations involving products of divisible sequences, extending known finiteness results from factorial-related equations to more general forms.

Contribution

It proves finiteness results for solutions of equations where a polynomial equals a product of divisible sequences, generalizing previous factorial-based findings.

Findings

Finiteness of solutions for polynomial equations involving divisible sequences.

Extension of Brocard-Ramanujan type results to broader classes of equations.

New conditions under which solutions are finite.

Abstract

Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for equations of the form $P(x)=\prod_{i=1}^rH_{n_i}$, where $P(x)$ is a polynomial and $H_{n_i}$ are divisible sequences.