Je\'{s}manowicz' conjecture and Fermat numbers

TL;DR

This paper proves Jeśmanowicz' conjecture for specific Pythagorean triples involving Fermat numbers, confirming the conjecture's validity in these cases.

Contribution

It establishes the truth of Jeśmanowicz' conjecture for a new class of Pythagorean triples related to Fermat numbers.

Findings

Jeśmanowicz' conjecture holds for triples involving Fermat numbers.

The paper verifies the conjecture for the specific triples (F_k-2, 2^{2^{k-1}+1}, F_k).

Supports the conjecture's validity in special cases involving Fermat numbers.

Abstract

Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}.$ In 1956, Je\'{s}manowicz 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 $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be a Fermat number. In this paper, we show that Je\'{s}manowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.