Bibliographic record
Abstract
In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by y2 = x3 + Ax + B, where A and B are integers, has no rational points of finite order if A is sufficiently large relative to B (at least if one assumes the abc Conjecture of Masser and Oesterlé). In the present article we show, perhaps surprisingly, that the rational torsion on the above curve is also quite restricted if B is sufficiently large relative to A. In particular, we demonstrate that for any ε > 0 there is a constant cε such that if A and B are integers satisfying |B| > cε |A|6+ε, then the elliptic curve defined above has no rational torsion points, other than a possible point of order 2 (again making use of the abc Conjecture in some cases). We then extend this by proving similar results for elliptic curves admitting non-trivial ℚ-isogenies, elliptic curves written in other forms, and elliptic curves over certain number fields. Curiously, the results on isogenies lead to two unexpected irrationality measures for certain algebraic numbers.
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How this classification was reachedexpand
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 | 0.001 |
| 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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".