Gain Bounds for Diagonal Superelliptic Equations under the Strong ABC Conjecture

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 $B{y}^n=A{x}^n+k$ 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.