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Claim. Let $f\in\mathbb{Z}[x]$ be irreducible of degree $k\ge3$. Hooley proved that the number of positive integers $n\le x$ with $f(n)$ $(k-1)$-power-free is asymptotic to $c_{f,k-1}\,x$, where

$$c_{f,k-1}=\prod_p\left(1-\frac{\rho_f(p^{k-1})}{p^{k-1}}\right)$$

and $\rho_f(q)$ counts the residues $a$ modulo $q$ with $f(a)\equiv0\pmod q$: the asymptotic Ricci had proved at exponents $k$ and above, now at the exponent $k-1$ at which Erdős 1953 had proved infinitude. This is what the first question of Problem 978 asks for: the $n\ge1$ with $f(n)$ $(k-1)$-power-free have positive natural density exactly when $c_{f,k-1}>0$, that is, when no prime $p$ has $p^{k-1}$ dividing every value of $f$. Under the question's hypotheses the constant is positive: an irreducible element of $\mathbb{Z}[x]$ is primitive, the fixed divisor of a primitive polynomial of degree $k$ divides $k!$, and $v_p(k!)\ge k-1$ holds only for $p=2$ with $k$ a power of $2$, the case the statement excludes (and Erdős's example $k!\bigl(\binom xk+1\bigr)$ shows that it occurs). The sign of the leading coefficient plays no role, since power-freeness is a property of $|f(n)|$.

Covers. The first question of Problem 978 (the part k_minus_1_density), answered yes: for irreducible $f\in\mathbb{Z}[x]$ of degree $k\ge3$ with $k$ not a power of $2$, the set of $n\ge1$ with $f(n)$ $(k-1)$-power-free has positive natural density $c_{f,k-1}$. Not covered: the second and third questions, at exponent $k-2$, which OpenAI's density theorem settles.

Source. C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21--26, DOI 10.1112/S002557930000797X. The paper is not held in the library. The statement above follows the site's remark, the survey in the release manuscript carded as OpenAI 2026, whose introduction cites Erdős 1953 for infinitude and Hooley 1967 for the asymptotic at exponent $k-1$, and the introduction of Heath-Brown, Power-free values of polynomials, Quart. J. Math. 64 (2013), 177--188 (arXiv:1103.2028v1), which states the asymptotic $N_{f,k}(x)\sim C(f,k)\,x$ with the Euler-product constant and records that Hooley obtained it for $k=d-1$, $d\ge3$. The paper appeared in volume 14, issue 1, dated June 1967, so this page is dated the first of June 1967.

Acceptance. Refereed: the paper appeared in Mathematika, a journal. The curator of erdosproblems.com, T. F. Bloom, states in the problem's remark that Hooley settled the first question, with a precise asymptotic for the number of such $n\le x$; since the site labels the problem OPEN (page last edited 31 March 2026), that remark is commentary and is not counted as review. Nothing on this page is independently reviewed by this project.

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