Short effective intervals containing primes in arithmetic progressions and the seven cubes problem
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
For any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> ϵ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\epsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and any non-exceptional modulus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q\ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we prove that, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> large enough ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x greater-than-or-equal-to alpha Subscript epsilon Baseline log squared q"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ≥ </mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ϵ </mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> </mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\ge \alpha _{\epsilon }\log ^2 q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), the interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket e Superscript x Baseline comma e Superscript x plus epsilon Baseline right-bracket"> <mml:semantics> <mml:mrow> <mml:mo>[</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>x</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi> ϵ </mml:mi> </mml:mrow> </mml:msup> <mml:mo>]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left [ e^x,e^{x+\epsilon }\right ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in any of the arithmetic progressions modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We apply this result to establish that every integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> larger than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis 71 000 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>71</mml:mn> <mml:mspace width="thinmathspace"/> <mml:mn>000</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp (71\,000)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sum of seven cubes.
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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