Problem Statement
Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\geq 1$) such that the leading coefficient of $f$ is positive.
Does the set of integers $n\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?
Are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?
In particular, does\[n^4+2\]represent infinitely many squarefree numbers?
Does the set of integers $n\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?
Are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?
In particular, does\[n^4+2\]represent infinitely many squarefree numbers?
Categories:
Number Theory
Progress
Erdős [Er53] proved there are infinitely many $n$ for which $f(n)$ is $(k-1)$-power-free, except for possibly when $k=2^l$, when it may happen that $2^{l-1}\mid f(n)$ for all $n$.Hooley [Ho67] settled the first question, in fact providing a precise asymptotic for the number of such $n\leq x$.
Heath-Brown [He06] proved the answer to the second question is yes when $k\geq 10$, and Browning [Br11] extended this to $k\geq 9$ (in fact establishing an asymptotic formula for the number of such $n$).
In [Er65b] Erdős mentions the question of whether $2^n\pm 1$ represents infinitely many $k$th power-free integers, or $n!\pm 1$, but that these are 'intractable at present'. (See also [936].)
Source: erdosproblems.com/978 | Last verified: January 19, 2026