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Bibliographic record
Abstract
The infamous <italic>twin prime conjecture</italic> states that there are infinitely many pairs of distinct primes which differ by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there are infinitely many pairs of distinct primes which differ by no more than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B equals 70000000"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mn>70000000</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">B = 70000000</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover, a co-operative team, <italic>Polymath8</italic> , collaborating only online, had been able to lower the value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4680"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4680</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{4680}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="600"> <mml:semantics> <mml:mn>600</mml:mn> <mml:annotation encoding="application/x-tex">600</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript m"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there are infinitely many intervals of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript m"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> containing at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct primes. We will also prove this much stronger result herein, even showing that one can take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript m Baseline equals e Superscript 8 m plus 5"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>8</mml:mn> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B_m=e^{8m+5}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . If Zhang’s method is combined with the Maynard–Tao setup, then it appears that the bound can be further reduced to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="246"> <mml:semantics> <mml:mn>246</mml:mn> <mml:annotation encoding="application/x-tex">246</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . If all of these techniques could be pushed to their limit, then we would obtain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it