Small prime solutions of quadratic equations II
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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="b 1 comma ellipsis comma b 5"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">b_1, \ldots , b_5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be non-zero integers and <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> any integer. Suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b 1 plus midline-horizontal-ellipsis plus b 5 identical-to n left-parenthesis mod 24 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo> ≡ </mml:mo> <mml:mi>n</mml:mi> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:mn>24</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">b_1+\cdots +b_5 \equiv n \pmod {24}</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="left-parenthesis b Subscript i Baseline comma b Subscript j Baseline right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">(b_i,b_j)=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i greater-than j less-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>j</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq i > j \leq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper we prove that (i) if the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Subscript j"> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">b_j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are not all of the same sign, then the above quadratic equation has prime solutions satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript j Baseline much-less-than StartRoot StartAbsoluteValue n EndAbsoluteValue EndRoot plus max left-brace StartAbsoluteValue b Subscript j Baseline EndAbsoluteValue right-brace Superscript 25 slash 2 plus epsilon Baseline semicolon"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo> ≪ </mml:mo> <mml:msqrt> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:msqrt> <mml:mo>+</mml:mo> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>25</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <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">p_j\ll \sqrt {|n|}+ \max \{|b_j|\}^{25/2+\varepsilon };</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and (ii) if all the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Subscript j"> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:annotation encod
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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