A note on the exceptional solutions of Jeśmanowicz’ conjecture concerning primitive Pythagorean triples

Abstract

Let $$(m,\ n)$$ be fixed positive integers such that $$m>n,\ \gcd (m,\ n)=1$$ and $$ mn\equiv 0 \pmod 2$$ . Then the triple $$(m^2-n^2,\ 2mn,\ m^2+n^2)$$ is called a primitive Pythagorean triple. In 1956, Jeśmanowicz (Wiadom Math 1(2):196–202, 1955/1956 ) conjectured that the equation $$(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$$ has only the positive integer solution $$(x,\ y,\ z)=(2,\ 2,\ 2)$$ . This problem is not solved yet. A solution $$(x,\ y,\ z)$$ of this equation is called exceptional if $$(x,\ y,\ z)\ne (2,\ 2,\ 2)$$ . In this paper, using Baker’s method, we prove that if $$m>\max \{10^{127550},\ n^{5127},\ n^{(\log n)^2}\}$$ , then the above equation has no exception solutions $$(x,\ y,\ z)$$ with $$ x\equiv y\equiv 0 \pmod 2 $$ . By this conclusion, we can deduce that if m, n satisfy the above condition, and $$ m\equiv 3 \pmod 4 $$ or $$ n\equiv 3 \pmod 4 $$ , then Jeśmanowicz’ conjecture is true.