Short··0:20·Number theory
Jacobsthal's Function: OpenAI Claims the Quadratic Bound
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About this video
Pick k primes: how many consecutive whole numbers can all be divisible by at least one of them before a number slips through? Jacobsthal asked whether that longest run h(k) is always at most a constant times k², and the best bound since Iwaniec in 1978 was k²(log k)². OpenAI claims h(k) ≪ k²/(log log 3k)² for every set of k primes and every interval, answering Jacobsthal's question; this is OpenAI's preprint, not yet peer reviewed. The towers are real computed runs for the first 1 to 9 primes (for example 9440 to 9460 for 2, 3, 5, 7, 11, 13).
Paper (OpenAI, catalogue result 021):
- A quadratic bound for Jacobsthal's function: github.com/openai/math/blob/main/preprints/A-quadratic-bound-…
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/021
About the problem
Number theory·OpenAI Math·Result 021
A quadratic bound for Jacobsthal’s function
Answers Jacobsthal's quadratic-bound question: every interval of Ck2 consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss. (Lean)
Topics
Jacobsthal functionJacobsthal's functionJacobsthal quadratic boundcoprime integersprime gapsprimorialsieve theoryIwaniecconsecutive integerscovering systemsnumber theoryanalytic number theoryprimesOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
