Erdős Problem #978

declared status 'open'. Formalized: yes. 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? Current best: 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 $n$). Prize: no. Tags: number theory.

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