Claims

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2025_09_16_van_doorn: An elementary argument credited to Wouter van Doorn in the site's commentary that f(N) <= (25/28 + o(1))N for Problem 301, by counting disjoint dilates of {2, 3, 4, 6, 12}; it exists only as site commentary and is pending.

2026_05_27_wang: An unrefereed manuscript of 27 May 2026 by Xinjun Wang claims f(N) <= (667/806 + o(1))N for the extremal function of Problem 301, by a divisor-dilation argument on the divisors of 720 with a finite certificate.

2026_07_30_della_pietra: A partial proof claim of 30 July 2026 by Donald Della Pietra, AI-assisted, that f(N) >= (1/2 + epsilon)N for an absolute epsilon and all large N, which would answer the particular question of Problem 301 in the negative.

2026_07_30_della_pietra_upper: An upper bound f(N) <= (15437/19344 + o(1))N for Problem 301, announced on 30 July 2026 in Donald Della Pietra's repository with an exact certificate over the divisors of 5040; AI-assisted, unrefereed, formalized only at M = 60.

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