On the Exceptional Solutions of Jeśmanowicz’ Conjecture

Let (a, b, c) be a primitive Pythagorean triple. Set a = m2 − n2, b = 2mn , and c = m2 + n2 with m and n positive coprime integers, m n and m 􀀁≡ n (mod 2). A famous conjecture of Je´smanowicz asserts that the only positive integer solution to the Diophantine equation ax + by = cz is (x, y, z) = (2, 2, 2). A solution (x, y, z) 􀀁= (2, 2, 2) of this equation is called an exceptional solution. In this note, we will prove that for any n 0 there exists an explicit constant c(n) such that if m c(n), the above equation has no exceptional solution when all x,y and z are even. Our result improves that of Fu and Yang (Period Math Hung 81(2):275–283, 2020). As an application, we will show that if 4 || m and m c(n), then Je´smanowicz’ conjecture holds.