Bibliographic record
Abstract
The infamous twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by 2 2 . Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound B B such that there are infinitely many pairs of distinct primes which differ by no more than B B . This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable. Zhang even showed that one can take B = 70000000 B = 70000000 . Moreover, a co-operative team, Polymath8 , collaborating only online, had been able to lower the value of B B to 4680 {4680} . They had not only been more careful in several difficult arguments in Zhang’s original paper, they had also developed Zhang’s techniques to be both more powerful and to allow a much simpler proof (and this forms the basis for the proof presented herein). In November 2013, inspired by Zhang’s extraordinary breakthrough, James Maynard dramatically slashed this bound to 600 600 , by a substantially easier method. Both Maynard and Terry Tao, who had independently developed the same idea, were able to extend their proofs to show that for any given integer m ≥ 1 m\geq 1 there exists a bound B m B_m such that there are infinitely many intervals of length B m B_m containing at least m m distinct primes. We will also prove this much stronger result herein, even showing that one can take B m = e 8 m + 5 B_m=e^{8m+5} . If Zhang’s method is combined with the Maynard–Tao setup, then it appears that the bound can be further reduced to 246 246 . If all of these techniques could be pushed to their limit, then we would obtain
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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 | 0.002 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.015 | 0.003 |
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".