Short··0:21·Number theory
Squarefree Values of Quartics: OpenAI Claims the Conjecture
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About this video
A number is squarefree when no prime square divides it. The conjecture says an irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the density predicted by the Euler product ∏(1 − ρ(p²)/p²). OpenAI claims a proof, plus (d−2)-power-free values for degrees 4 to 8, which together with Browning's theorem covers every degree d ≥ 4; this is OpenAI's preprint, not yet peer reviewed. In the animation each n up to 1500 is a bead for f(n) = n⁴ + n + 6: beads drop to the tier of the smallest prime square dividing f(n), and the 620 survivors (41.3%) match the Euler product's 41.3%.
Paper (OpenAI, catalogue result 020):
- Squarefree values of quartics and power-free values of polynomials: github.com/openai/math/blob/main/preprints/Squarefree-values-…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/020
About the problem
Number theory·OpenAI Math·Result 020
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)-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)
Topics
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