December 5, 2025Open Access

Paper Content, Height, and S-integers in a+b=c: Unconditional Core and Beal as a Corollary

Key Points

Strict bounds exist for coprime triples in equations with exponents ≥ 3, and verification is simplified to local checks.
Major outcomes include exclusions for infinite coprime families and the coprime Beal case, with all constants deemed effective.
In-depth analysis features p-adic methods and low bounds in logs, streamlining proof processes for mathematical conditions.
Impacts highlight effective strategies for bounding solutions in complex numerical forms, advancing mathematical theory.

Abstract

Unconditional content–height inequality with absolute gain for coprime triples a + b = c: log rad (abc) ≤ (1 − δ) (h (a) + h (b) + h (c) ) + C, with effective constants. The proof blends Archimedean lower bounds for linear forms in logs (Baker–Wüstholz/Matveev) and p-adic/Yu methods plus effective finiteness of S-integral solutions. Consequences include strict bounds for Aˣ + Bʸ = Cᶻ (x, y, z ≥ 3, gcd (A, B) = 1), excluding infinite coprime families and reducing verification to a finite, locally checkable set; a complete exclusion scheme for the coprime Beal case, with all constants effective.

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