Claims

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2025_11_20_van_doorn: Van Doorn's Pell construction, completed by Tao with Gemini Deepthink, makes the product of the powerful parts of three consecutive integers exceed c n^2 log n infinitely often, so <<_k n^2 fails for k at least 3; pending.

2026_06_10_hughes: Scott Hughes's Lean development that the product of the powerful parts of three consecutive integers exceeds any multiple of n squared log n, by the Pell construction over many primes at once; no public paper, pending.

2026_06_10_hughes_conditional: Scott Hughes's conditional theorem that, under a radical lower bound implied by the abc conjecture, the product of the powerful parts of k consecutive integers is at most a constant times n^{2+eps}; no public paper, pending.

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