Claims

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1938_10_01_rankin: Rankin's 1938 lower bound for the largest prime gap below X, with the constant 1/3, answers the question yes for every C below 1/3; refereed in the Journal of the London Mathematical Society.

2014_08_20_ford_green_konyagin_tao: Proves that the largest prime gap below X exceeds Rankin's 1938 scale by a factor tending to infinity, so the question's bound holds for every constant; refereed in the Annals and credited by the site's curator.

2014_08_21_maynard: Proves that the ratio of the prime gap to Rankin's scale has infinite limit superior, so the question's bound holds for every constant; refereed in the Annals and credited by the site's curator beside the independent 2014 proof.

2014_12_16_ford_green_konyagin_maynard_tao: Proves that the largest prime gap below X is at least a constant times log X log log X log log log log X over log log log X, a factor log log log X beyond the question's bound; refereed in JAMS and credited by the curator.

2026_08_26_alexeev: Two Lean modules in Boris Alexeev's lean-proofs repository prove the question for every C > 0 and the 2018 five-author bound, naming no informal author; claimed, not built by this corpus.

2026_08_26_dottedcalculator: A manuscript written by an AI system and posted by the forum user DottedCalculator proves prime gaps of order log n log log n over the fourth iterated logarithm, beyond the question's bound; credited by the curator.

2026_09_03_openai: A short OpenAI note attributing its proof to GPT 6 Astra claims the largest prime gap below X exceeds a constant times log X (log log X)^2 times the fourth iterated logarithm over the square of the third; pending.

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