CONTEMPORARY METHODS FOR SOLVING DIOPHANTINE EQUATIONS MICHAEL BENNETT (UNIVERSITY OF BRITISH COLUMBIA)
Bibliographic record
Abstract
The topic of this summer school was Diophantine equations, which are among the oldest studied mathematical objects. A Diophantine equation is an equation where admissible solutions are restricted to the rationals or the integers, or appropriate mathematical generalizations of such objects. The equations themselves tend to be polynomial, exponential, or a mixture of both, where variables in the exponents are usually restricted to (positive) integers. A characteristic example is the equation that is central to Fermat’s Last Theorem, x n + y n = z n, with x, y, z ∈ Z and n ∈ {3, 4,...}. Because Diophantine equations concern themselves with objects so fundamental to mathematics, they tend to arise whenever one uses the mathematical language to formulate problems or theories. This supplies a dual motivation to the field. On the one hand, there is an interest to understand theoretically the set of solutions to the equations and its relationship to the geometric objects defined by the equations. On the other hand, there is a demand for practical methods that, given an explicit equation, provide a complete and explicit description of the set of solutions. In recent years, a combination of development of general theory, computational tools and computational techniques has greatly improved our ability to
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.005 |
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".