Problem 978

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claims/: The 6 claim pages of Problem 978, one per claimant's result; the problem's standing derives from them.

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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?

Formulation. The preamble excludes degrees $k=2^l$, yet the third question, introduced as a particular case, concerns $n^4+2$, of degree $4=2^2$; a thread comment of September 2025 raised the inconsistency. The exclusion matters only for the first question, at exponent $k-1$, where Erdős 1953's exceptional polynomials occur: for $k=2^l$ the polynomial $k!\bigl(\binom xk+1\bigr)$ has $2^{k-1}$ dividing every value. This page reads the exclusion as belonging to the first question and the third question as the quartic case of the second question's exponent $k-2$, squarefreeness at $k=4$. The claims recorded below for the second and third questions need neither the exclusion nor the sign condition on the leading coefficient, so they answer either reading.

Status. Proved, departing from the site's label OPEN (page last edited 31 March 2026; proof-claims tab empty on 2026-10-06). The frontmatter lists the three questions as the problem's parts; each is settled by an accepted partial claim page, Hooley's asymptotic for the first and OpenAI's density theorem for the second and third, and the frontmatter standing is derived from Hooley's and OpenAI's pages together, since between them they settle every part.

Source. erdosproblems.com/978, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #978, https://www.erdosproblems.com/978.

References.

• [Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139-150.
• [Er53] Erdős, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416-425.
• [Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
• [He06] Heath-Brown, D. R., Counting rational points on algebraic varieties. (2006), 51-95.
• [Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21-26.

Formalization. Statement in formal-conjectures (revision of 2026-09-18): the file states the three questions as erdos_978.parts.i, erdos_978.parts.ii and erdos_978.parts.iii, the first marked solved and the other two open, all without proof. It also holds three variants: erdos_978.variants.sub_one, Erdős 1953's infinitude of $(k-1)$-power-free values, and erdos_978.variants.sub_two, Browning's case of degree at least $9$ for the second question, both marked solved without proof; and erdos_978.variants.allow_fixed_divisors, the second question with the local condition imposed at exponent $k-1$ instead of $k-2$, answered false with a formal proof of the counterexample described below in a fork. It is a statement file, not a formalization of any solution.

Current assessment

The site's formulation asks three questions about an irreducible $f\in\mathbb{Z}[x]$ of degree $k>2$ with positive leading coefficient, excluding $k=2^l$: whether the $n$ with $f(n)$ $(k-1)$-power-free have positive density; whether, for $k>3$ and $f$ without a fixed prime $(k-2)$-th power divisor, $f(n)$ is $(k-2)$-power-free infinitely often; and whether $n^4+2$ is squarefree infinitely often. The site's remarks record the literature as follows. Erdős [Er53] proved that $f(n)$ is $(k-1)$-power-free for infinitely many $n$, except in the case $k=2^l$ with $2^{k-1}$ dividing every value, which does occur, for instance at $f(x)=k!\left(\binom{x}{k}+1\right)$; the library card is Erdős 1953. Hooley [Ho67] answered the first question with an asymptotic count, recorded on its claim page. For the second question, Heath-Brown [He06] gave the answer yes for $k\geq 10$ and Browning [Br11] for $k\geq 9$, with an asymptotic formula, both under the local condition that no prime $p$ has $p^{k-2}$ dividing every value, which Erdős left implicit; they are recorded on Heath-Brown's claim page (claimed, since the lecture-notes volume is not documented as refereed) and Browning's claim page (accepted on the journal publication). Erdős [Er65b] also mentions whether $2^n\pm 1$ or $n!\pm 1$ represent infinitely many $k$-th-power-free integers and calls those questions intractable.

The remaining range $4\leq k\leq 8$ of the second question, and with it the third question, are settled by the accepted partial claim OpenAI 2026, whose source card is OpenAI 2026, squarefree values of quartics: for every irreducible $f$ of degree $k\geq 4$ meeting the local condition, the $(k-2)$-power-free values occur on a set of positive natural density given by the Euler product of local factors, and in particular $n^4+2$ is squarefree for a positive-density set of $n$. The claim is accepted on formalized evidence alone: this corpus built the release's Lean declaration and checked its axioms. No refereed publication or outside review of the manuscript is recorded. The first question is settled by Hooley's asymptotic, recorded as the accepted partial claim Hooley 1967 on refereed evidence (the journal publication; the curator's remark crediting Hooley is commentary on a problem the site labels OPEN); that page states why the Euler-product constant is positive under the question's hypotheses. The three questions are the problem's parts, each settled by an accepted partial claim, so the derived standing is solved with the claim proved, every answer being yes. The site's last edit (31 March 2026) predates the release.

Two earlier preprints claim the third question and are recorded as pending partial claims. Carella's page records Squarefree Values Of Polynomials (arXiv:2310.16952, math.GM, version 1 of 2023-10-25), whose Theorems 1.1--1.3 claim asymptotics for the squarefree values of $n^4+1$, $n^4+2$ and $n^4-2n^2+2$; its abstract names only quartic polynomials. Zapata Ceballos and Jalalvand's page records On the Squarefree Values of Degree-$2q$ Polynomials (arXiv:2608.10335, version 1 of 2026-08-11), whose Theorem 1.1 claims positive density of squarefree values for a monic irreducible integer polynomial of degree $2q$, $q$ prime, with a squarefree fixed divisor and whose root field contains a Galois subfield of degree $q$; the release notes that this includes $n^4+1$, and $x^4+2$ meets these hypotheses, since its root field contains $\mathbb{Q}(\sqrt{-2})$ and its fixed divisor is $1$. The release manuscript cites both preprints and uses neither. Both claims agree with the accepted answer and leave the derived standing unchanged.

The site's discussion thread (16 comments as of 2026-10-07) records the formulation's history. On 31 March 2026 a comment reported that a DeepMind prover agent had found a counterexample to the second question as then worded, without the local condition: the sextic $X^6+33X^5+21X^4+63X^3+18X^2+24X+48$ is congruent to $X(X-1)\cdots(X-5)$ modulo $16$, so $16$ divides every value and no value is $4$-power-free. The curator added the local condition the same day, and the statement above carries it. A comment of the same day reports that two AI systems, named there as ChatGPT 5.4 and Gemini 3.1 Pro, claim a positive answer for $n^4+2$ under the abc conjecture; no manuscript is linked, and a result conditional on abc decides nothing, so it gets no claim page. Earlier comments (September 2025 to March 2026) explain the exclusion of $k=2^l$ through Erdős's examples, note that Bunyakovsky's conjecture would make $n^4+2$ prime, hence squarefree, infinitely often, and correct the statement's wording.

Search scope, 2026-10-07: the site's problem page, remarks, discussion thread and proof-claims tab (empty on 2026-10-06); the release manuscript, its Lean catalog and the corpus's verification record; the arXiv records of the two preprints above; the formal-conjectures file at the pinned commit. Not searched: zbMATH, MathSciNet and X.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1953_arithmetical_properties_polynomials
• erdos_1953_arithmetical_properties_polynomials / remarks_p425
• erdos_1953_arithmetical_properties_polynomials / theorem
• openai_2026_squarefree_values_quartics_power_free_values_polynomials
• openai_2026_squarefree_values_quartics_power_free_values_polynomials / corollary_1_2
• openai_2026_squarefree_values_quartics_power_free_values_polynomials / theorem_1_1
• erdos_1965_recent_advances_current_problems_number_theory