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Abstract

For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical
systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$
$n\ge 1.$ We apply this to improve the lower bound on the number of distinct
quadratic fields of the form $\mathbb{Q}(\sqrt{F_P(n)})$ in short intervals
$M\le n\le M+H$ previously due to Cilleruelo, Luca, Quir\'{o}s and Shparlinski.
As a second application, we estimate the average number of missing values of
$F_P(n)(\text{mod}\ p)$ for special families of polynomials, generalizing
previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.