Title: 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 $$P_{m,n}:=f(1)f(2)\ldots f(m)$$. We prove that if $$m > \max \{10^{12},4^{n+1}\}$$, then there exists a prime divisor p of $$P_{m,n}$$ such that $$\mathrm{ord}_{p}(P_{m,n} )\le n\cdot 2^{n-1}$$. For $$n=2$$, we establish that for every positive integer m, there exists a prime divisor p of $$P_{m,2}$$ such that $$\mathrm{ord}_{p} (P_{m,2}) \le 4$$. Consequently, $$P_{m,2}$$ is never a fifth or higher power. This extends work of Cilleruelo [6] who studied the case $$n=1$$.