What question did this study set out to answer?

To rigorously prove the Beal Conjecture using algebraic methods and number theory principles.

May 30, 2026Open Access

化核理论下比尔猜想的证明

Key Points

To rigorously prove the Beal Conjecture using algebraic methods and number theory principles.
Constructed a proof based on nuclear decomposition and unique representation.
Analyzed the conditions of coprimality and parity of A, B, and C using 2-adic valuation and modular constraints.
Employed a descent argument to derive a contradiction in the fundamental assumptions.
Demonstrated that A and C must be identical, contradicting the assumption of their coprimality.
Establishes conditions under which A, B, and C must adhere to specific parity constraints.

Abstract

This paper presents a rigorous algebraic proof of the Beal Conjecture using the Nuclear Decomposition N=2i3mR−1N=2i3mR−1 (unique representation). Assuming a coprime solution Ax+By=CzAx+By=Cz with x,y,z>2x,y,z>2, we show that BB must be even and A,CA,C odd. Through 2-adic valuation analysis, modular constraints, and a descent argument, we prove that AA and CC must be identical, contradicting coprimality. The proof uses only elementary number theory, binomial expansions, and properties of 2-adic and 3-adic valuations, requiring no unproven conjectures.

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