\(2^x=y^2-1=\left(y-1\right)\left(y+1\right)\Rightarrow\left\{{}\begin{matrix}y-1=2^m\\y+1=2^n\end{matrix}\right.\) với \(\left\{{}\begin{matrix}m+n=x\\m< n\end{matrix}\right.\)
\(\Rightarrow2^n-2^m=2\Leftrightarrow2^m\left(2^{n-m}-1\right)=2.1\)
\(\Rightarrow\left\{{}\begin{matrix}2^m=2\\2^{n-m}-1=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}m=1\\n=2\end{matrix}\right.\) \(\Rightarrow x=3\Rightarrow y=3\)