A note on Jesmanowicz' conjecture concerning primitive Pythagorean triples

摘要

Let (a, b, c) be a primitive Pythagorean triple such that a = -u(2) - v(2), b = 2uv, c = u(2) - v(2), where u, v are positive integers satisfying u > v, gcd(u, v) = 1 and 2 vertical bar uv. In 1956, L. Jesmanowicz conjectured that the equation (an)(x)+(bn)(y) = (cn)(z) has only the positive integer solutions (x, y, z, n) = (2, 2, 2, m), where m is an arbitrary positive integer. A positive integer solution (x, y, z, n) of the equation is called exceptional if (x, y, z) not equal (2, 2, 2) and n > 1. In this paper we prove the following results: (i) The equation has no positive integer solutions (x, y, z, n) which satisfy x = y, y > z and n > 1. (ii) If (x, y, z, n) is an exceptional solution of the question, then either y > z > x or x > z > y. (iii) If u = 2(r), v = 2(r) 1, where r is a positive.integer, then the equation has no exceptional solutions (x, y, z, n) with y > z > x. In particular, if 2(r) 1 is an odd prime, then the equation has no exceptional solutions. The last result means Jegmanowicz conjecture is true in this case. (C) 2015 Elsevier Inc. All rights reserved.