Prime powers dividing products of consecutive integer values of $x^{2^n}+1$

Abstract

Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every positive integer $m$, there exists a prime divisor $p$ of $P_{m,2}$ such that ord$_{p} (P_{m,2}) \leq 4$. Consequently, $P_{m,2}$ is never a fifth or higher power. This extends work of Cilleruelo who studied the case $n=1$.