On Jesmanowicz' conjecture concerning primitive Pythagorean triples

JOURNAL OF NUMBER THEORY

卷:141

Article

Terai, Nobuhiro

关键词: Pythagorean triples;    Exponential Diophantine equations;    Generalized Fermat equations;    Linear forms in two logarithms;   

DOI  :  10.1016/j.jnt.2014.02.009

来源: Elsevier

【 摘 要 】

In 1956, Jesmanowicz conjectured that the exponential Diophantine equation (m(2) - n(2))(x) + (2mn)(y) = (m(2) + n(2))(x) has only the positive integer solution (x, y, z) = (2, 2, 2), where m and n are positive integers with m > n, gcd(m,n) = 1 and m not equivalent to n (mod 2). We show that if n = 2, then Jesmanowicz' conjecture is true. This is the first result that if n = 2, then the conjecture is true without any assumption on m. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free