Bibliographic record
Abstract
Numerous papers have studied the problem of determining upper bounds for the number of integer points on elliptic curves of the form [Formula: see text], and quartic curves of the form [Formula: see text]. Bounds for the number of integer solutions to such quartic equations typically depend on both of the coefficients [Formula: see text]. The purpose of this paper is to examine more closely how the number of integer points on such quartic curves seems to depend almost entirely on the number of prime factors of [Formula: see text]. This is done by focusing in on the problem of bounding the number of squares in certain recurrence sequences. In particular, using some arithmetic on elliptic curves, it is proved that infinitely many such sequences can have four squares, but that a fifth square remains elusive after extensive computation, suggesting that an absolute bound for the number of squares is more than likely, and also suggesting that the above assertion regarding the dependence on the number of prime factors of [Formula: see text].
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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 | 0.003 | 0.003 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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".