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Erdős problem 539

For ∣A∣=n|A| = n, how small can the cofactor set Q(A)={a/gcd⁡(a,b):a,b∈A}Q(A) = \{a / \gcd(a,b) : a, b \in A\} be? The answer is h(n)=n1/2+o(1)h(n) = n^{1/2 + o(1)}: a new upper bound h(n)≤n1/2exp⁡(O(log⁡n))h(n) \le n^{1/2} \exp(O(\sqrt{\log n})) matches the classical lower bound.

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FormalConjectures/ErdosProblems/

539.lean

Retained formal statement4 of 8

From [Er73]: The determination of lim⁡n→∞log⁡(h(n))log⁡(n) \lim_{n\to\infty}\frac{\log(h(n))}{\log(n)} will perhaps be not too difficult.

FormalConjectures/ErdosProblems/539.leanErdos539.erdos_539.variants.limit1 lineExact file
Filter.Tendsto (fun n => Real.log ↑(Erdos539.cofactorThreshold n) / Real.log ↑n) Filter.atTop sorry
OpenStatement only, no proof

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