A note on the Diophantine equation \((na)^x + (nb)^y = (nc)^z\)

Authors

  • M. J. Deng Hainan University

Keywords:

Diophantine equations, Pythagorean triples, Je\'smanowicz' conjecture

Abstract

Let \((a, b, c)\) be a primitive Pythagorean triple satisfying \(a^2 +b^2 = c^2.\) In 1956, Je\'smanowicz conjectured that for any given positive integer \(n\) the only solution of \((an)^x + (bn)^y = (cn)^z\) in positive integers is \(x = y = z = 2.\) In this paper, for the primitive Pythagorean triple \((a, b, c)= (4k^2 - 1, 4k , 4k^2 + 1)\) with \(k=2^s\) for some positive integer \(s\geq 0\), we prove the conjecture when \(n >1\) and certain divisibility conditions are satisfied. 10.1017/S000497271300066X

Published

2014-01-27

Issue

Section

Articles