Bibliographic record
Abstract
James Maynard has taken the analytic number theory world by storm in the last decade, proving several important and surprising theorems, resolving questions that had seemed far out of reach. He is perhaps best known for his work on small and large gaps between primes (which were discussed, hot off the press, in our 2015 Bulletin of the AMS article). In this article we will discuss two other Maynard breakthroughs: — Mersenne numbers take the form 2 n − 1 2^n-1 and so appear as 111 … 111 111\dots 111 in base 2, having no digit “ 0 0 ”. It is a famous conjecture that there are infinitely many such primes. More generally it was, until Maynard’s work, an open question as to whether there are infinitely many primes that miss any given digit, in any given base. We will discuss Maynard’s beautiful ideas that went into his 2019 partial resolution of this question. — In 1926, Khinchin gave remarkable conditions for when real numbers can usually be “well approximated” by infinitely many rationals. However Khinchin’s theorem regarded 1/2, 2/4, 3/6 as distinct rationals and so could not be easily modified to cope, say, with approximations by fractions with prime denominators. In 1941 Duffin and Schaeffer proposed an appropriate but significantly more general analogy involving approximation only by reduced fractions (which is much more useful). We will discuss its 2020 resolution by Maynard and Dimitris Koukoulopoulos.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.036 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.024 | 0.007 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".