A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples

• Main Authors: Van Thien Nguyen, Viet Kh. Nguyen, Pham Hung Quy
• Format: Article
• Language: English
• Published: Ptolemy Scientific Research Press 2021-03-01
• Series: Open Journal of Mathematical Sciences
• Subjects: diophantine equations; non-primitive pythagorean triples; jeśmanowicz conjecture.
• Online Access: https://pisrt.org/psr-press/journals/oms-vol-5-2021/a-note-on-jesmanowicz-conjecture-for-non-primitive-pythagorean-triples/

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. Jeśmanowicz 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 \((x,y,z)≠(2,2,2)\)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 Jeśmanowicz conjecture for all prime values \(u<100\)).