On the conjecture of Je\'smanowicz

TL;DR

This paper surveys 60 years of research on Je'smanowicz' conjecture and presents a new result proving the specific Diophantine equation has only the trivial solution.

Contribution

It provides a comprehensive review of past results and introduces a novel proof for a particular case of the conjecture.

Findings

The Diophantine equation has no solutions other than (2,2,2).

The survey summarizes key developments over 60 years.

The new proof advances understanding of the conjecture.

Abstract

We give a survey on some results covering the last 60 years concerning Je\'smanowicz' conjecture. Moreover, we conclude the survey with a new result by showing that the special Diophantine equation $$(20k)^x+(99k)^y=(101k)^z$$ has no solution other than $(x,y,z)=(2,2,2)$.