On the Exceptional Solutions of Jeśmanowicz’ Conjecture

Abstract

Let (a, b, c) be a primitive Pythagorean triple. Set $$a = m^2-n^2 $$ , $$b=2mn$$ , and $$c=m^2+n^2$$ with m and n positive coprime integers, $$ m>n $$ and $$m \not \equiv n \pmod 2 $$ . A famous conjecture of Jeśmanowicz asserts that the only positive integer solution to the Diophantine equation $$a^x+b^y=c^z$$ is $$(x,y,z)=(2,2,2).$$ A solution $$(x,y,z) \ne (2,2,2)$$ 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 $$4 \mid \!\mid m$$ and $$m > c(n) $$ , then Jeśmanowicz’ conjecture holds.