Ternary Diophantine Equations via Galois Representations and Modular Forms
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Full frame distilled prediction
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.
- Candidate categories
- none
- Consensus categories
- none
- Domain
- Candidate signal: noneConsensus signal: none
- Study design
- Candidate signal: Theoretical or conceptualConsensus signal: Theoretical or conceptual
- Genre
- Candidate signal: EmpiricalConsensus signal: none
- Teacher disagreement score
- 0.407
- Threshold uncertainty score
- 0.673
- Validation status
machine_predicted_unvalidated·codex-gemma-dda1882f352a
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| 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 | 0.001 |
| 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)
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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.
- Teacher spread
- 0.264 · how far apart the two teachers sit on this one work
- Validation status
score_only:v0-immature-baseline· verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it
Abstract
Abstract In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2 , based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A , B and C . We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.
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The record
- Venue
- Canadian Journal of Mathematics
- Topic
- Advanced Mathematical Identities
- Field
- Mathematics
- Canadian institutions
- University of British Columbia
- Funders
- not available
- Keywords
- MathematicsDiophantine equationGalois moduleTernary operationGalois groupPolynomialAlgebra over a fieldExponential functionPure mathematicsDiscrete mathematicsMathematical analysis
- Has abstract in OpenAlex
- yes