On the Extendibility of a D(4)-Pair of Pell Numbers
Bibliographic record
Abstract
A Diophantine m-tuple with property D(ℓ) is a set of m integers such that the product of any two integers plus ℓ results in a perfect square. This thesis establishes that a particular family of D(4) pairs of Pell numbers can be extended to a D(4) triple by exactly one Pell number. A similar result has been found for the Diophantine triples of Fibonacci numbers, a discussion of which is included in the first chapter of this thesis. This chapter finishes with a statement of the main result of my thesis, and the subsequent chapters discuss several topics in number theory which were used to prove the main result in chapter 5. Specifically, results about continued fractions, Pell-type equations, and linear forms in logarithms were used. These topics are the subjects of chapters 2, 3 and 4, which contain some history and discussions of the important results. The conclusion of this thesis discusses some possible generalizations.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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".