A note on Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. II

Abstract

Let ${(m^2 - n^2, 2mn, m^2 + n^2)}$ be a primitive Pythagorean triple such that m, n are positive integers with ${ \gcd (m,n)=1}$, ${m > n}$, ${m\not\equiv n\pmod{2}}$. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the exponential Diophantine equation ${(m^2-n^2)^x + (2mn)^y = (m^2+n^2)^z}$ is x =  y =  z =  2. Let ${(m,n)\equiv(u,v)\pmod{d}}$ denote ${m\equiv u\pmod{d}}$ and ${n\equiv v\pmod{d}}$. Using the theory of quartic residue character and elementary method, we first prove Jeśmanowicz’ conjecture in the following cases. (i) ${(m,n)\equiv(1,2)\pmod{4}}$. (ii) ${(m,n)\equiv(3,2)}$, ${(7,6)\pmod{8}}$ or ${(m,n)\equiv(3,6)}$, (7,2), (11,14), (15,10) ${(\mod{16})}$. (iii) ${(m,n)\equiv(3,14)}$, (7,10), (11,6), ${(15,2)\pmod{16}}$ and ${y > 1}$. Then, by using the above results, two lemmas that based on Laurent’s deep result and computer assistance, for ${n\equiv2\pmod{4}}$ with ${n < 600}$, we prove the conjecture without any assumption on m.