A note on Terai's conjecture concerning primitive Pythagorean triples

Abstract

Let $f,g$ be positive integers such that $f>g$, $\gcd(f,g)=1$ and $f\not\equiv g \pmod{2}$. In 1993, N. Terai conjectured that the equation $x^2+(f^2-g^2)^y=(f^2+g^2)^z$ has only one positive integer solution $(x,y,z)=(2fg,2,2)$. This is a problem that has not been solved yet. In this paper, using elementary number theory methods with some known results on higher Diophantine equations, we prove that if $f=2^rs$ and $g=1$, where $r,s$ are positive integers satisfying $2\nmid s$, $r\ge 2$ and $s<2^{r-1}$, then Terai's conjecture is true.