Jeśmanowicz' conjecture and Fermat numbers

Abstract

Let $a,b,c$ be relatively prime positive integers such that $a^2+b^2=c^2.$ In 1956, Jeśmanowicz conjectured that for any positive integer $n$, the only solution of $(an{)}^x+(bn{)}^y=(cn{)}^z$ in positive integers is $(x,y,z)=(2,2,2)$. Let $k\ge 1$ be an integer and $F_k=2^{2^k}+1$ be a Fermat number. In this paper, we show that Jeśmanowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.

we correct some mistakes in the first version and revised the title of paper