Gain Bounds for Diagonal Superelliptic Equations under the Strong ABC Conjecture
Karsten Müller
TL;DR
The paper develops a framework to bound the adapted power gain $G_p$ and approximation gain $G_a$ for coprime solutions of the diagonal superelliptic equation $B y^n = A x^n + k$ under a strong ABC conjecture. By deriving a purely structural lower bound for $G_a$ and combining it with a global ABC-quality bound, it establishes a uniform upper bound on $G_p$: $$G_p < q_{max}/G_{a,min}.$$ Under Ultra-Strong conjecture $q<1.5$, this yields concrete numerical caps (e.g., $G_p<3$ for $n=2$). The work further shows that for $k=1$, the condition $q>n/2$ excludes solutions for $n\ge 4$ when $q<2$, and illustrates sharpness with known ABC triples (Reyssat, de Weger) while highlighting the necessity of the non-triviality condition via the Nitaj example. These results illuminate how gain bounds constrain the exponents and constants in Diophantine diagonal superelliptic equations, contingent on unproven ABC conjectures.
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.