The Diophantine equation $P(x)=\stackrel{r}{\prod _{i=1}}{H}_{n_i}$

Abstract

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

9 pages, comments are welcome!