Jesmanowicz' conjecture revisited

Authors

  • M. Tang
  • Z.-J. Yang

Keywords:

Je\'{s}manowicz' conjecture, Diophantine equation

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 $(x,y,z)=(2,2,2)$. In this paper, we consider Je\'{s}manowicz' conjecture for Pythagorean triples $(a,b,c)$ if $a=c-2$ and $c$ is a Fermat prime. For example, we show that Je\'{s}manowicz' conjecture is true for $(a,b,c)=(3,4,5)$; $(15,8,17)$; $(255,32,257)$; $(65535,512,65537)$. 10.1017/S0004972713000038

Published

2013-09-27

Issue

Vol. 88 No. 3 (2013)

Section

Articles
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