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Claim. T. D. Browning, Power-free values of polynomials, Arch. Math. (Basel) 96 (2011), no. 2, 139--150. Let $f\in\mathbb{Z}[x]$ be irreducible of degree $d\ge3$, let $\rho_f(q)$ count the residues $a$ modulo $q$ with $f(a)\equiv0\pmod q$, and let $r\ge(3d+1)/4$. Then the number of $n\le x$ with $f(n)$ $r$-power-free is $c_{f,r}\,x+o(x)$, where $c_{f,r}=\prod_p\bigl(1-\rho_f(p^r)/p^r\bigr)$ is the Euler product of the local factors. The exponent $r=d-2$ satisfies $d-2\ge(3d+1)/4$ exactly when $d\ge9$. So every irreducible $f$ of degree at least $9$ with no prime $p$ such that $p^{d-2}$ divides every value takes $(d-2)$-power-free values on a set of $n$ of positive density $c_{f,d-2}$, hence infinitely often. The theorem needs neither the exclusion of degrees that are powers of $2$ nor the sign condition on the leading coefficient. Browning obtains the range by combining Heath-Brown's determinant method, which gave $r\ge(3d+2)/4$ (its claim page), with Salberger's global determinant estimates.

Covers. The second question of Problem 978 for every polynomial of degree $k\ge9$, answered yes, with a positive density in place of infinitude. Not covered: the degrees $4\le k\le8$ and the third question, both settled by OpenAI's density theorem, and the first question.

Read depth. The statement is checked against B. Z. Moroz's zbMATH review of the paper (Zbl 1252.11070), which states the asymptotic for irreducible $f$ of degree $d\ge3$ and $k\ge(3d+1)/4$ and the consequence for $d\ge9$ under the local condition $f(\mathbb{Z}/p^{d-2}\mathbb{Z})\ne\{0\}$ for every prime $p$; the review gives no theorem number, and the paper is not held. The sentence on the method follows Section 1.1 of the release manuscript carded as OpenAI 2026. The proof is not checked.

Depends on. No page of this wiki; the claim rests on the paper above.

Acceptance. Refereed: Arch. Math. (Basel) 96 (2011), 139--150. The site's curator credits Browning with the case $k\ge9$ in the problem's remarks, but the site labels the problem OPEN, so that credit is commentary and is not counted as review.