TY  - JOUR
T1  - A note on Terai's conjecture concerning primitive Pythagorean triples
AU  - Soydan, Gökhan
AU  - Le, Maohua
PY  - 2021
DA  - August
DO  - 10.15672/hujms.795889
JF  - Hacettepe Journal of Mathematics and Statistics
PB  - Hacettepe University
WT  - DergiPark
SN  - 2651-477X
SP  - 911
EP  - 917
VL  - 50
IS  - 4
LA  - en
AB  - 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.
KW  - polynomial-exponential Diophantine equation
KW  - generalized Ramanujan-Nagell equation
KW  - primitive Pythagorean triple
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UR  - https://doi.org/10.15672/hujms.795889
L1  - https://dergipark.org.tr/en/download/article-file/1294071
ER  -