OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Answers Jacobsthal's quadratic-bound question: every interval of Ck^2 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.
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Let h(k) be the least integer such that every interval of h(k) consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove h(k)\ll k^2/(loglog(3k))^2, giving an affirmative answer to Jacobsthal's quadratic-bound question.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
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