Corrigenda and addition to “Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100000000000”
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Bibliographic record
Abstract
An error in the program for verifying the Ankeny-Artin-Chowla (AAC) conjecture is reported. As a result, in the case of primes <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> which are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="identical-to 5 mod 8"> <mml:semantics> <mml:mrow> <mml:mo> ≡ </mml:mo> <mml:mn>5</mml:mn> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>8</mml:mn> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\equiv 5\bmod {8}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the AAC conjecture has been verified using a <italic>different</italic> multiple of the regulator of the quadratic field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q left-parenthesis StartRoot p EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>p</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}(\sqrt {p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> than was meant. However, since <italic>any</italic> multiple of this regulator is suitable for this purpose, provided that it is smaller than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="8 p"> <mml:semantics> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">8p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the main result that the AAC conjecture is true for all the primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="identical-to 1 mod 4"> <mml:semantics> <mml:mrow> <mml:mo> ≡ </mml:mo> <mml:mn>1</mml:mn> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\equiv 1\bmod {4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than 10 Superscript 11"> <mml:semantics> <mml:mrow> <mml:mo>></mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">>10^{11}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , remains valid. As an addition, we have verified the AAC conjecture for all the primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="identical-to 1 mod 4"> <mml:semantics> <mml:mrow> <mml:mo> ≡ </mml:mo> <mml:mn>1</mml:mn> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\equiv 1\bmod {4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10 Superscript 11"> <mml:semantics> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">10^{11}</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="2 times 10 Superscript 11"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> × </mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">2\times 10^{11}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with the corrected program.
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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) | 0.000 | 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