In:
Open Journal of Mathematical Sciences, Ptolemy Scientific Research Press, Vol. 5, No. 1 ( 2021-12-31), p. 115-127
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 \(u 〈 100\).
Type of Medium:
Online Resource
ISSN:
2616-4906
,
2523-0212
DOI:
10.30538/oms2021.0150
Language:
Unknown
Publisher:
Ptolemy Scientific Research Press
Publication Date:
2021
detail.hit.zdb_id:
2956263-6