A note on Jeśmanowicz' conjecture for non-primitive Pythagorean triples release_tohsmb37i5gi5fxa6gbvpetqoi

by Van Thien Nguyen, Department of Mathematics, Hoa Lac High Tech Park, FPT University, Hanoi, Vietnam, Viet Kh. Nguyen, Pham Hung Quy, Department of Mathematics and Information Assurance, Hoa Lac High Tech Park, FPT University, Hanoi, Vietnam, Department of Mathematics, Hoa Lac High Tech Park, FPT University, Hanoi, Vietnam

Published in Open Journal of Mathematical Sciences by Ptolemy Scientific Research Press.

2021   p115-127

Abstract

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), the Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\). In this connection we call a positive integer solution \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case \(v=2,\ u\) is an odd prime. As an application we show the truth of the Jesmanowicz conjecture for all prime values \(u < 100\).
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