On the conjecture of Je\'smanowicz

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 $(x,y,z)=(2,2,2)$.