On Jeśmanowicz' conjecture concerning primitive Pythagorean triples

In 1956, Jeśmanowicz conjectured that the exponential Diophantine equation (m2−n2)x+(2mn)y=(m2+n2)z(m2−n2)x+(2mn)y=(m2+n2)z has only the positive integer solution (x,y,z)=(2,2,2)(x,y,z)=(2,2,2), where m and n   are positive integers with m>nm>n, gcd(m,n)=1gcd(m,n)=1 and m≢n(mod2). We show that if n=2n=2, then Jeśmanowicz' conjecture is true. This is the first result that if n=2n=2, then the conjecture is true without any assumption on m.