Short··0:20·Number theory
Patterson's Conjecture on Cubic Gauss Sums: OpenAI Claims a Proof
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About this video
Every Eisenstein prime gives a cubic Gauss sum, a point on the unit circle, and their angles look random. Patterson conjectured that they still don't cancel: summed over primes of norm up to X they grow like X^(5/6)/log X, with an explicit constant. OpenAI claims an unconditional proof of this first-moment asymptotic over all primary primes; the animation shows all 6,028 real Gauss sums up to norm 60,000, whose sum is 990 against the main term's 877. This is OpenAI's preprint, not yet peer reviewed.
Papers (OpenAI, catalogue result 023):
- An unconditional first moment for cubic Gauss sums: github.com/openai/math/blob/main/preprints/An-unconditional-f…
The animation illustrates the result; it is not a proof.
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Every result explained: gradientdamage.com/p/023
About the problem
Number theory·OpenAI Math·Result 023
Patterson's first moment for cubic Gauss sums
Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order X5/6/log X. Every fixed nonzero prime-angle Fourier mode has smaller order. (Lean)
Topics
cubic Gauss sumsPatterson conjectureGauss sumKummer conjectureHeath-BrownEisenstein integersEisenstein primescubic residueexponential sumsanalytic number theorynumber theorymetaplectic formsOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
