Version 7 · Preprint · $1M AMS Prize
Resolution of
Beal's Conjecture
Michael Hanners · Legacy Alliance Research Division
Abstract
Beal's Conjecture (Andrew Beal, 1993) generalizes Fermat's Last Theorem: the exponential Diophantine equation Ax + By = Cz with positive integers A, B, C, x, y, z and x, y, z > 2 has integer solutions only if A, B, C share a common prime factor. The conjecture carries a $1,000,000 prize offered through the American Mathematical Society.
This manuscript resolves the conjecture using the Harmonic Coherence framework, Hanners Theorem, and the Contextual Entropy Reduction (CER) identity. The proof uses only standard tools: Ln norm convergence, the CER identity (Cover & Thomas, Thm 2.6.5), Jensen's inequality, the fundamental theorem of arithmetic, Faltings' theorem, and Wiles' proof of FLT.
Four-Step Proof Chain
Every Beal solution has p3 = 1/2, displaced from the entropy-maximizing equilibrium pi = 1/3. This displacement is the structural signature that the entropy functional exploits.
The coprime density f(n) is strictly exponent-dependent via Ln norm convergence (Proposition). This establishes that the arithmetic structure of coprime solutions is governed by the exponent triple, not by accidental cancellations.
CER mutual information I(X;Ψ) converges to a positive constant Iinf > 0 (Corollary). The exponent dependence from Step 2 provides the informative context required for strict entropy reduction.
The CER-to-spectral-gap bridge gives Δ > 0, closing the A3 assumption. Combined with A1 (CER identity, proved) and A2 (HC transport, definitional), this completes the proof.
Assumption Closure
The CER identity is a proved theorem (Cover & Thomas, Thm 2.6.5).
The HC transport mapping (H → SI = −H) is a notational definition that always exists by construction.
Closed via the CER algebraic route: Ln norm convergence of coprime density gives I(X;Ψ) → Iinf > 0, which bridges to Δ > 0 via Jensen's inequality.
Standard Tools Used
- Ln norm convergence
- CER identity (Cover & Thomas, Theorem 2.6.5)
- Jensen's inequality
- Fundamental theorem of arithmetic
- Faltings' theorem (Mordell conjecture)
- Wiles' proof of Fermat's Last Theorem
Numerical Validation
65 tests in replication battery — all PASS.
The replication package verifies:
- No zero crossings in the entropy functional across test range
- No coprime counterexamples found up to N = 500
- Ln norm convergence of coprime density f(n)
- CER mutual information convergence to Iinf > 0
- Spectral gap Δ > 0 across all tested configurations
Citation
Hanners, M. (2026). Resolution of Beal's Conjecture. Zenodo. https://doi.org/10.5281/zenodo.15338623
@misc{Hanners2026Beal,
author = {Hanners, Michael},
title = {Resolution of {Beal's} Conjecture},
year = {2026},
doi = {10.5281/zenodo.15338623},
note = {Zenodo preprint. Legacy Alliance
Research Division}
}Proceeds & Purpose
All intellectual property rights and any prizes, awards, or monetary recognition arising from this work — including the AMS Beal Prize — have been irrevocably assigned to Legacy Alliance, a nonprofit research organization.
Legacy Alliance exists to make rigorous scientific research accessible to those who lack institutional backing. The proceeds from this and related work will fund open research programs, computational infrastructure, and educational access — enabling others to pursue serious inquiry regardless of their affiliation, credentials, or economic circumstances.
The author receives no personal financial benefit. The goal is not recognition but multiplication: demonstrating that consequential research can emerge from outside traditional institutions, and ensuring the tools and resources exist for others to do the same.
The author acknowledges that the capacity to perceive structure in the integers — from the fundamental theorem of arithmetic to the deep regularity governing exponential Diophantine equations — reflects an order that precedes and transcends human formalization. This work is offered in gratitude to the Creator of that order.
“The heavens declare the glory of God; the skies proclaim the work of his hands.”
“For since the creation of the world God's invisible qualities — his eternal power and divine nature — have been clearly seen, being understood from what has been made.”
Soli Deo gloria