Rational Points in Arithmetic Progressions on <i>y</i><sup>2</sup> = <i>x<sup>n</sup></i> + <i>k</i>
Bibliographic record
Abstract
Abstract Let C be a hyperelliptic curve given by the equation y 2 = f ( x ) for f ∈ ℤ[ x ] without multiple roots. We say that points P i = ( x i , y i ) ∈ C (ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers x i for i = 1, 2, … , m are in arithmetic progression. In this paper we show that there exists a polynomial k ∈ ℤ[ t ] with the property that on the elliptic curve ε ′ : y 2 = x 3 + k ( t ) (defined over the field ℚ( t )) we can find four points in arithmetic progression that are independent in the group of all ℚ( t )-rational points on the curve Ε′. In particular this result generalizes earlier results of Lee and Vélez. We also show that if n ∈ ℕ is odd, then there are infinitely many k 's with the property that on curves y 2 = x n + k there are four rational points in arithmetic progressions. In the case when n is even we can find infinitely many k 's such that on curves y 2 = x n + k there are six rational points in arithmetic progression.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.044 | 0.017 |
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; both teacher heads agree on what is shown here.
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".