About the project
The long-term goal of this project is to build a reproducible research framework for the generalized Beal equation
A^x + B^y = C^z,
where the exponents are greater than two and the solution is primitive.
The full generalized problem is far too large to approach as a single computation, so we started with one difficult signature:
A^3 + B^5 = C^7.
The (3,5,7) case became our first end-to-end test of the method. It allowed us to develop the arithmetic tools, computational pipeline and proof-auditing process needed for future work on other exponent triples.
Instead of asking an AI model to produce a convincing-looking proof, i built a workflow in which mathematical ideas must be converted into exact computations, saved certificates and explicit proof dependencies. The system proposes steps, runs symbolic and algebraic calculations, checks finite cases, records failed approaches and flags circular reasoning.
For the (3,5,7) signature, the project includes:
- construction of the relevant Frey curve;
- local arithmetic and conductor calculations;
- ray class group computation;
- enumeration and filtering of 360 characters;
- residual irreducibility analysis;
- exact Brandt-module and Hecke-operator computations;
- reproducible certificates for the main finite calculations;
- a logical audit of the proof structure.
The remaining obstacle is no longer a large computer search. It is a narrow theoretical step involving a weight-two Hilbert Eisenstein argument and a final Jacquet–Langlands/Brandt bridge. We document this gap openly and do not claim that the full theorem has already been proved.
How it's built
The project combines:
- Python and PARI/GP for exact arithmetic;
- SageMath-style algebraic workflows;
- custom scripts for local calculations and Hecke filtering;
- persistent computational certificates;
- automated checks for proof dependencies and circular arguments;
- an AI agent used as a research coordinator in custom AI runtime qOrchestra, built by myself earlier
One of the most important design choices was to preserve failed ideas and corrections. During the project, several arguments initially looked plausible but later failed because of a missing hypothesis or circular use of irreducibility. Keeping those traces made the final research state much more trustworthy.
What i learned
I learned that advanced mathematical work with AI becomes much more reliable when the model is connected to deterministic tools and forced to leave an auditable trail.
The AI was useful for organizing a large proof search, translating mathematical questions into computational tasks and finding weak links in the argument. However, it could not replace exact computation, source verification or a careful dependency audit.
I also learned that finishing the finite search does not automatically finish the theorem. In this project, the computational part is largely complete, but a small theoretical bridge still carries most of the remaining mathematical difficulty.
The repository clearly separates:
- verified computations;
- closed lemmas;
- heuristic expectations;
- incomplete theorem bridges;
- invalid or circular shortcuts.
It also turns a broad generalized problem into a concrete, reproducible research program, beginning with the (3,5,7) signature.
What's next
The immediate goal is to close the final survivor (90,0) using the Eisenstein route, complete the theorem-level Jacquet–Langlands/Brandt justification and produce a clean final manuscript for the (3,5,7) case.
After that, the larger goal is to reuse the same architecture for additional exponent signatures and gradually identify which parts of the method can contribute to a broader treatment of the generalized Beal problem.
The project is therefore both a serious study of one specific generalized Fermat signature and an experiment in building a reusable AI-assisted workflow for modern arithmetic geometry.
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