Gain Bounds for Diagonal Superelliptic Equations under the Strong ABC Conjecture

TL;DR

This paper develops a new theoretical framework for bounding the gains of solutions to diagonal superelliptic equations, linking these bounds to the Strong ABC conjecture and validating results with known ABC triples.

Contribution

It introduces a structural lower bound for approximation gain and establishes bounds on power gain under the Strong ABC conjecture, connecting these to high ABC-quality solutions.

Findings

Power gain is bounded by the maximum ABC quality divided by the minimal approximation gain.

Solutions for $n \,\ge\, 4$ are excluded under $q < 2$ due to structural density.

Validation with known ABC triples confirms the sharpness of the bounds.

Abstract

We establish a novel framework for bounding the adapted power gain $G_p$ and approximation gain $G_a$ of coprime integer solutions to the generalized diagonal superelliptic equation $By^n = Ax^n + k$ with $x, y \ge 2$. By first deriving a purely structural lower bound for $G_a$, we demonstrate that these equations are inherently predisposed to high ABC-qualities ($q = G_a \cdot G_p$). Combined with the Strong ABC conjecture ($q < q_{max}$), we prove that the power gain is uniformly bounded by $G_p < q_{max}/G_{a,min}$, providing a theoretical foundation for the numerical observation $G_p < 3$ for $n=2$ under the Ultra-Strong conjecture ($q < 1.5$). Specifically, we show that for $k=1$, the structural density forces $q > n/2$, which excludes solutions for $n \ge 4$ under $q < 2$. We validate our theoretical bounds using high-quality ABC triples, specifically analyzing the Reyssat (1987), de Weger (1985), and Nitaj (1993) cases to demonstrate the sharpness of the structural approximation gain.