Roth's Theorem implies a Weakened Version of the ABC Conjecture for Special Cases

TL;DR

This paper shows that Roth's theorem can imply a weaker, non-effective form of the ABC Conjecture for specific cases involving algebraic roots, expanding understanding of their interrelation.

Contribution

It establishes a novel implication from Roth's theorem to a weakened version of the ABC Conjecture for special algebraic cases, which was previously unexplored.

Findings

Roth's theorem implies a weakened ABC Conjecture in certain algebraic root cases

Uses explicit formulas for continued fractions of algebraic numbers

Provides new insights into the relationship between Roth's theorem and the ABC Conjecture

Abstract

Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem implies a weakened non-effective version of the ABC Conjecture in certain cases relating to roots.