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 a2+b2=c2. 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)=(4k2−1,4k,4k2+1) with k=2s for some positive integer s≥0, we prove the conjecture when n>1 and certain divisibility conditions are satisfied. 10.1017/S000497271300066X

Published

2014-01-27

Issue

Vol. 89 No. 2 (2014)

Section

Articles
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