On the approximation gain for abc-triples

TL;DR

This paper generalizes the concept of approximation gain from specific abc-triples related to surd convergents to all abc-triples, supported by extensive numerical analysis.

Contribution

It introduces a generalized framework for approximation gain applicable to all abc-triples, expanding the scope of previous work.

Findings

Numerical computations support the proposed generalization.

The paper identifies various variants of the approximation gain concept.

Results suggest potential new insights into abc-triples and their properties.

Abstract

The concept of approximation gain was introduced recently by M\"uller and Taktikos for some abc-triples related to convergents of surds, where there is a relatively large gap between min{a,b,c} and max{a,b,c}. This note proposes a generalization of the concept to all abc-triples, with several variants. Extensive numerical computations are provided.