On the Beal Conjecture: A Constructive Proof via Differential Algebraic Closure and Height

This paper presents the first complete constructive proof of the Beal Conjecture within a unified framework of differential algebraic closure and differential algebraic height. The Beal Conjecture asserts that if positive integers A,B,C,x,y,z satisfy Ax + By = Cz with x,y,z ≥ 3, then A,B,C must share a common prime factor. The core of our proof lies in the introduction and rigorous definition of two novel concepts: differential algebraic closure and differential algebraic height.First, for any positive integer n, we construct a universal differential algebraic closure Kn, a differentially closed field extension obtained by formally adjoining abstract generators representing “critical points”, formal derivatives, roots of unity, and n-th roots of rational functions. For a given degree-n polynomial f, we prove the existence of a specialization homomorphism ιf : Kn → Lf mapping the universal closure to a differentially closed extension Lf of its coefficient field, wherein the roots of f acquire explicit representations. This construction fundamentally transcends the classical solvability-by-radicals paradigm, yet remains fully compatible with the Abel-Ruffini Theorem, as Lf generally does not reside within the traditional radical closure.Second, we define the differential algebraic height hDA(α) of a complex number α as the infimum of representation complexities over all possible differential algebraic expressions for α, simultaneously accounting for the arithmetic height (classical Weil height) of coefficients and the nested depth of radical and differential operations. This new invariant links the arithmetic properties of numbers with the complexity of their representation under extended differential operations.The proof path for the Beal Conjecture is as follows: Assuming a primitive counterexample (i.e., gcd(A,B,C) = 1), we associate it with a semi-stable elliptic curve Ex,y,z A,B,C. On one hand, the differential algebraic representation of its j-invariant (derived via a cubic equation) yields a lower bound for hDA(j), controlled by the logarithm of max(Ax,By,Cz). On the other hand, by the Modularity Theorem, the curve is modular, and its j-invariant admits a representation via modular functions through differential algebraic means, yielding an upper bound for hDA(j) controlled by the logarithm of the curve’s minimal discriminant ∆min(E). When x,y,z ≥ 3, these two inequalities become irreconcilable, thus proving the non-existence of primitive solutions. Beyond proving the Beal Conjecture, this paper establishes a novel framework connecting differential algebra, classical arithmetic geometry, and the theory of modular forms, providing powerful new tools for handling exponential Diophantine equations.