Corrigenda and addition to “Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100000000000”
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 p p which are ≡ 5 mod 8 \equiv 5\bmod {8} , the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field Q ( p ) \mathbb {Q}(\sqrt {p}) than was meant. However, since any multiple of this regulator is suitable for this purpose, provided that it is smaller than 8 p 8p , the main result that the AAC conjecture is true for all the primes ≡ 1 mod 4 \equiv 1\bmod {4} which are > 10 11 >10^{11} , remains valid. As an addition, we have verified the AAC conjecture for all the primes ≡ 1 mod 4 \equiv 1\bmod {4} between 10 11 10^{11} and 2 × 10 11 2\times 10^{11} , with the corrected program.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.017 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.262 | 0.109 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".