Improved lower bound for large gaps between consecutive primes
A new sieving construction raises the record lower bound on how large the gap between consecutive primes becomes infinitely often, the first advance on Erdős Problem #4 since 2018.
- Lab
- Independent
- Model
- GPT-5.6 Sol
- Field
- Mathematics
- Date
- 2026-08-25
- Human collaborators
- DottedCalculator, Boris Alexeev
- Problem posed
- 1955 · open 71 yrs
Sources
Original work
Announcement
Independent commentary
What was found
Writing G(T) for the largest gap between consecutive primes below T, the record since Ford, Green, Konyagin, Maynard and Tao (2018) has been G(T) much greater than log T log_2 T log_4 T / log_3 T, where log_k is the k-fold iterated logarithm. The manuscript claims G(T) much greater than log T log_2 T / log_4 T, a gain of log_3 T / (log_4 T)^2, together with Y(X) much greater than X log X / log_3 X for the covering problem behind it. What is new is narrow and specific: the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt, while the hypergraph covering theorem and the Maynard weight are carried over unchanged from the 2018 paper. Ben Green, a co-author of the bound being beaten, describes the sieving procedure as different from the Erdős-Rankin one that underpinned every bound on the problem since 1938.
Novelty check
Erdős Problem #4 is tracked at erdosproblems.com/4 and carries a $10,000 prize; the prize question itself was settled in 2016 and the record bound since is Ford, Green, Konyagin, Maynard and Tao (2018). Green, who holds that record, read the manuscript and compared the new sieve against Erdős-Rankin (1938) directly, which fixes both what was known and what is added. No intervening record appears in the erdosproblems.com record for the problem or in the vibemathed entry. The result is a new construction, not a retrieval.
Caveats and known objections
The manuscript has no human author: its title block and PDF metadata both name GPT 5.6 Sol, there is no acknowledgements section, and no human is credited anywhere in it. That is the argument for grading autonomy as autonomous; ai-led is taken instead because a human posed the problem, iterated on the output and asked for the argument to be made self-contained, and the weaker grade wins when the reading is arguable. The submitter, who is pseudonymous and wishes to stay so, states plainly that the mathematics was not theirs: 'I am not familiar with sieve theory.' Verification rests on Green becoming, in his own words, more or less convinced the argument is correct after consulting Tao and Maynard; that is expert reading, not refereeing. One attribution discrepancy is recorded rather than smoothed over: the manuscript says GPT 5.6 Sol, while the erdosproblems.com claim was filed as GPT 5.6 Pro. Boris Alexeev's Lean development transcribes the manuscript downstream rather than producing it, and this registry has not confirmed that it builds, so the grade stays at independent rather than formal. Readers on the r/math thread note that several later sections reproduce the 2018 paper with no new content, and Green calls the exposition horrific.
Also recorded at
vibemathedtilted-residue-class-construction-for-long-prime-free-intervals
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
Flag this for triage
Signals order the review queue and nothing else. They are never published, and they never move a grade: that takes a citation.
Entry history (1 event)
- AddedEntered the registry graded Independently checked and AI-led.
Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.
Graded independently checked for verification and ai-led for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Improved lower bound for large gaps between consecutive primes. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-25-erdos-4-prime-gapsBibTeX
@misc{whataifound-independent-2026-gaps,
title = {Improved lower bound for large gaps between consecutive primes},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Independently checked. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-25-erdos-4-prime-gaps}
}