Let f(t)∈Z[t] be a separable polynomial such that f(t)≠ a(t)b(t)^2 and operatornamedeg f≥ 1. Let mu denote the Möbius function and let rho_f(m) denote the number of residue classes modulo m satisfying f(t)equiv 0 pmod m. Squarefree-value conjecture. ∑_n≤ xmu^2(f(n))=x∏_p≥ 2(1-fracrho_f(p^2)p^2)+o(x). This predicts an asymptotic density for squarefree values of a broad class of separable polynomials. The surrounding discussion identifies the result as matching conditional consequences of the abc conjecture, but the supplied text does not establish its resolution.

Squarefree-value conjecture for polynomial values

About 3 years old · traced to 2023

Let $f(t)\in\mathbb{Z}[t]$ be a separable polynomial such that $f(t)\ne a(t)b(t)^2$ and $\operatorname{deg} f\geq 1$ . Let $\mu$ denote the Möbius function and let $\rho_f(m)$ denote the number of residue classes modulo $m$ satisfying $f(t)\equiv 0\pmod m$ . Squarefree-value conjecture.

\sum_{n\leq x}\mu^2(f(n))=x\prod_{p\geq 2}\left(1-\frac{\rho_f(p^2)}{p^2}\right)+o(x).

This predicts an asymptotic density for squarefree values of a broad class of separable polynomials. The surrounding discussion identifies the result as matching conditional consequences of the $abc$ conjecture, but the supplied text does not establish its resolution.

References

Primary source

N. A. Carella, “Squarefree Values Of Polynomials”, arXiv:2310.16952 (2023).

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims the predicted positive Euler-product density for squarefree values of every irreducible integer quartic with no fixed prime-square divisor, at positive integer inputs. This addresses the irreducible quartic subcase of the broader squarefree-value conjecture on this page. Its higher-degree results concern (d-2)-power-free values, rather than squarefree values of arbitrary degree.See full solutionHide full solution

Claimed by OpenAI.

The manuscript claims the predicted positive Euler-product density for squarefree values of every irreducible integer quartic with no fixed prime-square divisor, at positive integer inputs. This addresses the irreducible quartic subcase of the broader squarefree-value conjecture on this page. Its higher-degree results concern (d-2)-power-free values, rather than squarefree values of arbitrary degree.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf

OpenAI-020-01-Squarefree-values-of-quartics-and-power-free-values-of-polynomials.pdf676,811 bytesOpen