{"schema":"vela.problem-packet.v0.1","problem":978,"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?If $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$, then 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?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}