A form of refined Roth's theorem and its application to the $abc$-conjecture

TL;DR

This paper presents a refined version of Roth's theorem and applies it to prove a special case of the $abc$-conjecture, advancing understanding in Diophantine approximation and number theory.

Contribution

It introduces a new form of Roth's theorem and demonstrates its application to a specific case of the $abc$-conjecture, which is a significant problem in number theory.

Findings

A new form of refined Roth's theorem established.

Proved a special case of the $abc$-conjecture using this refinement.

Provides insights into Diophantine approximation and number theory.

Abstract

In this paper, we give a form of refined Roth's theorem. As an application, we prove a special case of the $abc$-conjecture.