Congruences like Atkin’s for the partition function
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis upper Q cubed script l n plus beta right-parenthesis identical-to 0 left-parenthesis mod script l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≡ </mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p( Q^3 \ell n+\beta )\equiv 0\pmod \ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are prime and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5 less-than-or-equal-to script l less-than-or-equal-to 31"> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi> ℓ </mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>31</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">5\leq \ell \leq 31</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; these lie in two natural families distinguished by the square class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 minus 24 beta left-parenthesis mod script l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mn>24</mml:mn> <mml:mi> β </mml:mi> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1-24\beta \pmod \ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In recent decades much work has been done to understand congruences of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis upper Q Superscript m Baseline script l n plus beta right-parenthesis identical-to 0 left-parenthesis mod script l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≡ </mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(Q^m\ell n+\beta )\equiv 0\pmod \ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is now known that there are many such congruences when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , that such congruences are scarce (if they exist at all) when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 1 comma 2"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow>
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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