A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples

Let (a,b,c) be a primitive Pythagorean triple such that a=u2−v2a=u2−v2, b=2uvb=2uv, c=u2+v2c=u2+v2, where u, v   are positive integers satisfying u>vu>v, gcd⁡(u,v)=1 and 2|uv2|uv. In 1956, L. Jeśmanowicz conjectured that the equation (an)x+(bn)y=(cn)z(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)≠(2,2,2) and n>1n>1. In this paper we prove the following results: (i) The equation has no positive integer solutions (x,y,z,n) which satisfy x=yx=y, y>zy>z and n>1n>1. (ii) If (x,y,z,n) is an exceptional solution of the question, then either y>z>xy>z>x or x>z>yx>z>y. (iii) If u=2ru=2r, v=2r−1v=2r−1, where r   is a positive integer, then the equation has no exceptional solutions (x,y,z,n) with y>z>xy>z>x. In particular, if 2r−12r−1 is an odd prime, then the equation has no exceptional solutions. The last result means Jeśmanowicz conjecture is true in this case.