OPEN
This is open, and cannot be resolved with a finite computation.
Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\geq 1$).
Does the set of integers $n$ 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?
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].)
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This page was last edited 19 October 2025.
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T. F. Bloom, Erdős Problem #978, https://www.erdosproblems.com/978, accessed 2026-01-16
