On the conjecture of Jeśmanowicz

Abstract

We give a survey on some results covering the last 60 years concerning Jeśmanowicz' 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 $(x,y,z)=(2,2,2)$.

24 pages, to appear, International Journal of Applied Mathematics & Statistics (2017)