Original scientific paper

https://doi.org/10.3336/gm.54.2.02

An open problem on Jeśmanowicz' conjecture concerning primitive Pythagorean triples

Hai Yang ; School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi, 710048, P.R. China
Ruiqin Fu ; School of Science, Xi'an Shiyou University, Xi'an, Shaanxi, 710065, P.R. China

Abstract

Let \(m>31\) be an even integer with \(\gcd(m,31)=1\). In this paper, using some elementary methods, we prove that the equation \((m^2-31^2)^x+(62m)^y=(m^2+31^2)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\). This result resolves an open problem raised by T. Miyazaki ({\em Acta Arith.} 186 (2018), 1--36) about Je\'smanowicz' conjecture concerning primitive Pythagorean triples.

Keywords

Ternary purely exponential Diophantine equation; Jeśmanowicz' conjecture; primitive Pythagorean triple; elementary method

Hrčak ID:

229599

URI

https://hrcak.srce.hr/229599

Publication date:

11.12.2019.

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