It's well known that there are no non-constant polynomials with integer coefficients whose values at integer points are primes. Could this result be generalized to the case of prime powers?


The question is whether there exists a polynomial $p(x) \in \mathbb{Z}[x]$ with degree at least one such that for all $x \in \mathbb{Z}$ $|p(x)|$ is prime power.



Polynomial with prime powers values
With integer coefficients the answer is surely no: let $f(x)\in\mathbb{Z}[x]$ a degree $d$ polynomial, and let $a\in \mathbb{Z}$ be such that $f(a)\neq0,1,-1$. Then there exist a prime $p$ such that $f(a)=0 \mod p$. Now consider the sequence $x\_n=f(a+np)$. Since $x\_n=0\mod p$, by the assumption on the values of $f$, we must have $x\_n=p^{\alpha\_n}$ for some positive integer $\alpha\_n$. Moreover, since $x\_n$ is definitely strictly increasing (it is no loss of generality assuming the leading coefficient of $f$ is positive), then so is $\alpha\_n$. Therefore $p^{\alpha\_n}$ has at least an exponential behaviour. But $x\_n=O(n^d)$.


For rational coefficients I don't know, but I expect a negative answer in that case, too.


