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Bibliographic record
Abstract
The Cunningham project seeks to factor numbers of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Superscript n Baseline plus-or-minus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo> ± </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b^{n}\pm 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b equals 2 comma 3 comma ellipsis"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">b=2,3,\dots</mml:annotation> </mml:semantics> </mml:math> </inline-formula> small. One of the most useful techniques is <italic>Aurifeuillian Factorization</italic> whereby such a number is partially factored by replacing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Superscript n"> <mml:semantics> <mml:msup> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">b^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by a polynomial in such a way that polynomial factorization is possible. For example, by substituting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y equals 2 Superscript k"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">y=2^{k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into the polynomial factorization <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 y squared right-parenthesis squared plus 1 equals left-parenthesis 2 y squared minus 2 y plus 1 right-parenthesis left-parenthesis 2 y squared plus 2 y plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2y^{2})^{2}+1=(2y^{2}-2y+1)(2y^{2}+2y+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we can partially factor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript 4 k plus 2 Baseline plus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2^{4k+2}+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In 1962 Schinzel gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified <italic>all</italic> numbers that can be factored by such identities and we prove this if one accepts our definition of what “such an identity” is. We then develop our theme to similarly factor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis b Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(b^{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any given polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-for
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.001 | 0.001 |
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