Result 020, Number theory

Squarefree quartics and power-free polynomial values

Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the (d−2)(d-2)-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.

Lean formalization Proof

The bigger picture

Why it matters

A polynomial can produce numbers with repeated prime factors, but how often does it avoid them? This work claims an exact limiting frequency for squarefree values of eligible degree-four polynomials.

What changes?

The manuscript reports predicted positive density of squarefree values at positive integer inputs for irreducible quartics: degree-four integer polynomials not factorable over the rationals. Squarefree means divisible by no prime square; no prime square may divide every value. It also reports the corresponding density for values divisible by no prime power of exponent d minus two, for degrees d from four through eight, under the analogous condition. With Browning's higher-degree theorem, this latter claim covers every d at least four.

What does that help mathematicians do?

The density is an Euler product: multiply, over all primes, the fraction of input residue classes that avoid the forbidden prime power. The claimed result makes these local divisibility tests sufficient to determine a positive long-run proportion of acceptable values. Researchers could therefore deduce not just infinitely many squarefree values for eligible quartics, but their asymptotic frequency. It does not establish squarefree density in every higher degree.

Are there practical applications?

Its immediate value is foundational: it connects divisibility conditions checked one prime at a time with the distribution of polynomial values across the positive integers. The reported proof brings number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry into this counting problem.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Squarefree values of quartics and power-free values of polynomials

September 24, 2026 55 pages Main result formalized in Lean

We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding (d−2)(d-2)-power-free density in degrees 4≤d≤84\le d\le8. The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the (d−2)(d-2)-power-free density for every d ≥ 4.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026,
  author = {{OpenAI}},
  title = {{Squarefree values of quartics and power-free values of polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf}{OAI:Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/020.md.

Squarefree quartics and power-free polynomial values

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result proves positive-density power-free values for every integer polynomial ff irreducible over Q\mathbb Q of degree d≥4d\ge4. Put k=d−2k=d-2 and assume no prime kkth power divides every value of ff. Then the number of positive integers n≤Xn\le X for which f(n)f(n) is kk-free is cfX+of(X)c_fX+o_f(X), where cf>0c_f>0 is the convergent product of local factors. Negative values are allowed and zero is excluded. No monicity, primitivity, or coefficient-height restriction is imposed.

Comparator links

Result Comparator statement
Positive density of power-free polynomial values PowerFreeValues.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.