A note on Jeśmanowicz’ 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}\...
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doaj-f74694a2fb16471dbdecb4341d0c6cef2021-04-03T15:30:03ZengPtolemy Scientific Research PressOpen Journal of Mathematical Sciences2616-49062523-02122021-03-015111512710.30538/oms2021.0150A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triplesVan Thien Nguyen0Viet Kh. Nguyen1Pham Hung Quy2Department of Mathematics, Hoa Lac High Tech Park, FPT University, Hanoi, Vietnam.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.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 \(u<100\)).https://pisrt.org/psr-press/journals/oms-vol-5-2021/a-note-on-jesmanowicz-conjecture-for-non-primitive-pythagorean-triples/diophantine equationsnon-primitive pythagorean triplesjeśmanowicz conjecture. |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Van Thien Nguyen Viet Kh. Nguyen Pham Hung Quy |
spellingShingle |
Van Thien Nguyen Viet Kh. Nguyen Pham Hung Quy A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples Open Journal of Mathematical Sciences diophantine equations non-primitive pythagorean triples jeśmanowicz conjecture. |
author_facet |
Van Thien Nguyen Viet Kh. Nguyen Pham Hung Quy |
author_sort |
Van Thien Nguyen |
title |
A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples |
title_short |
A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples |
title_full |
A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples |
title_fullStr |
A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples |
title_full_unstemmed |
A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples |
title_sort |
note on jeśmanowicz’ conjecture for non-primitive pythagorean triples |
publisher |
Ptolemy Scientific Research Press |
series |
Open Journal of Mathematical Sciences |
issn |
2616-4906 2523-0212 |
publishDate |
2021-03-01 |
description |
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 \(u<100\)). |
topic |
diophantine equations non-primitive pythagorean triples jeśmanowicz conjecture. |
url |
https://pisrt.org/psr-press/journals/oms-vol-5-2021/a-note-on-jesmanowicz-conjecture-for-non-primitive-pythagorean-triples/ |
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