A note on Terai`s conjecture concerning primitive Pythagorean triples

Authors : Maohua LE, Gökhan SOYDAN

Pages : 911-917

Doi:10.15672/hujms.795889

View : 20 | Download : 4

Publication Date : 2021-08-06

Article Type : Research Paper

Abstract :Let $f,g$ be positive integers such that $f>g$, $\gcdinsert ignore into journalissuearticles values(f,g);=1$ and $f\not\equiv g \pmod{2}$. In 1993, N. Terai conjectured that the equation $x^2+insert ignore into journalissuearticles values(f^2-g^2);^y=insert ignore into journalissuearticles values(f^2+g^2);^z$ has only one positive integer solution $insert ignore into journalissuearticles values(x,y,z);=insert ignore into journalissuearticles values(2fg,2,2);$. 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.
Keywords : polynomial exponential Diophantine equation, generalized Ramanujan Nagell equation, primitive Pythagorean triple