The Non-Existence of Integer Solutions to The Quartic-Cubic Diophantine Equation Y3 + Xy = X4 + 4: A Complete Resolution Via Factorization and Modular Arithmetic
Bibliographic record
Abstract
We establish the complete non-existence of integer solutions to the Diophantine equation y3 + xy = x4 +4, thereby resolving an open problem in the classification of quartic- cubic Diophantine equations. Our proof employs a novel synthesis of classical techniques: we utilize Sophie Germain’s identity for the factorization of quartic forms, develop a com- prehensive greatest common divisor stratification, and apply systematic modular arithmetic obstructions combined with the unique factorization property in Z. The proof proceeds through an exhaustive case analysis based on d = gcd(x, y), where we show that d ∈ {1, 2, 4} is necessary, and then demonstrate that each case leads to a polynomial equation with no integer roots. We establish several auxiliary results on the coprimality structure of the factored forms and the impossibility of certain quartic polynomial equations over Z. Our methods extend beyond this specific equation, providing a template for attacking similar mixed-degree Diophantine problems. We complement our theoretical analysis with rigorous computational verification and propose several generalizations, connecting our re- sult to the broader landscape of Diophantine analysis, including connections to genus-1 curves and the study of integral points on algebraic varieties. The techniques developed herein contribute to the ongoing classification program for Diophantine equations of low degree and small height.
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 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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.005 | 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".