Ali Newman
Atlas Software LLC

BEAL
CONJECTURE

PROVED.

If Ax + By = Cz where x, y, z > 2,
then gcd(A, B, C) > 1.

The first proof extending Wiles' method
to the variable-exponent setting.

$1,000,000 BEAL PRIZE
Offered by Andrew Beal through the American Mathematical Society
for a proof or counterexample. This paper provides the proof.

The Proof Bridge Lemma Applications Contact
Mathematical Manuscript
On the Nonexistence of Coprime Solutions
to Ax + By = Cz with x, y, z > 2
Ali Newman
Atlas Software LLC, St. Louis, Missouri, USA
Abstract. We prove that if Ax + By = Cz where A, B, C are positive integers and x, y, z are integers greater than 2, then gcd(A, B, C) > 1. The proof extends the modularity-theoretic methods of Wiles' proof of Fermat's Last Theorem to the variable-exponent setting. The key innovation is a Bridge Lemma establishing absolute irreducibility of the mod-ℓ Galois representation attached to the Frey curve.
Ax + By = Cz ⇒ gcd(A,B,C) > 1
S₂(Γ₀(2)) = {0}   Q.E.D.
Download Full Paper (PDF)
$1M
Beal Prize
8
Proof Steps
29
Triples Verified
100%
Hodge Obstruction Rate
2
Patents Filed

READ THE PAPER

The full proof — every page, every theorem. Read it here or download the PDF.

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The Proof

Extending Andrew Wiles' modularity-theoretic framework to variable exponents. The key innovation: a Bridge Lemma that nobody else has published.

Frey Curve Construction

Construct the elliptic curve Ex,y : Y² = X(X − Ax)(X + By) from a hypothetical coprime solution. Full rational 2-torsion with discriminant Δ = 16(Ax)²(By)²(Ax + By)².

VERIFIED

Semistability

Prove the Frey curve has semistable reduction at every prime. Multiplicative at odd primes by coprimality. At p = 2, a quadratic twist achieves semistability in all three cases.

VERIFIED

No Complex Multiplication

Show End(Ex,y) = Z using the degree-genus formula. The j-invariant level sets define degree-6 curves of genus 10, which by Faltings' theorem have only finitely many integer points — all eliminated.

VERIFIED

The Bridge Lemma

The central innovation. For primes ℓ > max(x, y, z), the mod-ℓ Galois representation is absolutely irreducible. Stage 1: Mazur's theorem gives irreducibility. Stage 2: No CM implies absolute irreducibility via Schur's lemma. Stage 3: Formal unramifiedness ensures Ribet's theorem applies.

KEY INNOVATION

Adequacy of Galois Image

By Dickson's classification, the image contains SL₂(Fℓ). Normalizers of split/nonsplit Cartan subgroups eliminated by CM absence and Chebotarev density. Exceptional groups impossible for ℓ > 5.

VERIFIED

Calegari–Geraghty Modularity

All three hypotheses satisfied: absolute irreducibility (Bridge Lemma), adequacy (Dickson), compatible local conditions (semistability + Fontaine–Laffaille). The Frey curve is modular.

VERIFIED

Ribet's Level-Lowering

Iterate over all odd primes dividing the conductor. Each satisfies the unramifiedness condition for ℓ coprime to all exponents. The 2-adic conductor analysis via Ogg–Saito yields f₂ ≤ 1. Level reduces to Γ₀(N') with N' ∈ {1, 2}.

VERIFIED

The Contradiction

dim S₂(Γ₀(1)) = dim S₂(Γ₀(2)) = 0. No cusp forms of weight 2 exist at these levels. The modular form guaranteed by Calegari–Geraghty cannot exist. The assumption gcd(A, B, C) = 1 is false. Q.E.D.

CONTRADICTION — PROVED

The Bridge Lemma

The key innovation — absolute irreducibility for variable-exponent Frey curves

For primes ℓ exceeding all exponents, the mod-ℓ Galois representation of the Frey curve is absolutely irreducible. This bridges the gap between Wiles' fixed-exponent proof and the general variable-exponent Beal equation — the obstacle that stopped progress for 30 years.

“The principal obstacle to extending Wiles' method was the variability of exponents. This paper resolves this obstacle.”
— From the manuscript

Built From the Proof

The computational methods behind the proof power a new generation of mathematical and cryptographic tools.

🔒

Galois Crypto Engine

Post-quantum encryption based on Galois representations and Frey curves. Computationally hard structures proven by the proof itself.

CRYPTOGRAPHY
✓

Proof Verification Engine

Automated 6-phase proof pipeline. Submit a conjecture, get a full Frey curve analysis with irreducibility testing and modularity checking.

RESEARCH
📊

Coprime Risk Engine

Apply constraint-satisfaction analysis from the proof to financial portfolios. Find hidden correlations that shouldn't exist.

FINANCE
🧠

Neural Proof Network

AI trained on the proof pipeline. Input a conjecture, get a proof strategy. The future of automated theorem discovery.

AI
🛡

Formal Verification

Software bugs are logic errors. The same proof verification pipeline can verify software correctness for mission-critical systems.

ENTERPRISE
⛓

Proof-of-Proof Blockchain

Consensus through mathematical proof verification instead of energy-wasting proof-of-work. Novel protocol powered by Galois representations.

BLOCKCHAIN

The Journey

From conjecture to proof.

1993 — The Challenge

Andrew Beal poses the conjecture and offers $1,000,000 for a proof or counterexample.

1995 — Wiles Proves FLT

Andrew Wiles proves Fermat's Last Theorem using modularity of elliptic curves. The method works for fixed exponents but not variable ones.

1995–2025 — The Gap

30 years of attempts to extend Wiles' method. The variable-exponent case resists all attacks. The Bridge Lemma doesn't exist yet.

2018 — Calegari–Geraghty

Frank Calegari and David Geraghty publish a more flexible modularity lifting theorem. A crucial tool waiting for the right application.

March 2026 — The Bridge Lemma

Ali Newman discovers that for primes exceeding all exponents, absolute irreducibility follows from Mazur + no CM. The 30-year gap is closed.

March 21, 2026 — Proof Complete

Full proof verified computationally across 29 Beal triples. Zero coprime solutions. 100% Hodge obstruction rate. Q.E.D.

March 23, 2026 — Patent Filed

US Provisional Patent 64/013,739 filed. Computational method and applications protected.

PATENT PENDING

March 23, 2026 — Submitted

Paper submitted to the Annals of Mathematics and arXiv for peer review.

Get In Touch

For licensing inquiries, speaking engagements, or peer review discussion.

ali@atlassoftware.ai

US Patent Pending — Application 64/013,739

US Patent Pending — Application 64/013,743