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Claim. D. R. Heath-Brown, Counting rational points on algebraic varieties, in Analytic Number Theory, Lecture Notes in Math. 1891, Springer, 2006, 51--95, Theorem 16. Let $f\in\mathbb{Z}[x]$ be irreducible of degree $d$ and let $r\ge(3d+2)/4$. Then the number of $n\le x$ with $f(n)$ $r$-power-free is $c_{f,r}\,x+o(x)$, with $c_{f,r}=\prod_p\bigl(1-\rho_f(p^r)/p^r\bigr)$ the Euler product of the local factors. The exponent $r=d-2$ satisfies $d-2\ge(3d+2)/4$ exactly when $d\ge10$, so every irreducible $f$ of degree at least $10$ with no prime $p$ such that $p^{d-2}$ divides every value takes $(d-2)$-power-free values on a set of positive density. The proof applies the affine determinant method. The volume's record gives only the year, so this page is dated the first of January 2006.

Covers. The second question of Problem 978 for every polynomial of degree $k\ge10$, answered yes, with a positive density in place of infinitude. Browning's refereed theorem (its claim page) extends the range to $k\ge9$.

Read depth. The statement is taken from the descriptions of Theorem 16 in the introduction of Heath-Brown, Power-free values of polynomials, Quart. J. Math. 64 (2013) (arXiv:1103.2028v1, p. 1), and in Section 1.1 of the release manuscript carded as OpenAI 2026. The chapter is not held, and B. Z. Moroz's zbMATH review of it (Zbl 1152.11027) says only that the lectures treat power-free values of polynomials, improving on known results, without stating the theorem or its range. The chapter's proof is not checked here.

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

Standing. The chapter appeared in a lecture-notes volume, and no record documents that the volume was refereed, so the page lists no refereed evidence. The site's curator credits the result in the problem's remarks, but the site labels the problem OPEN, so that credit is commentary and is not counted as review. The claim stays claimed.