Consider equation $2x^2+1=y^2$. We have: Let $0< x_0 < y_0$ be an integer solution of $2x^2+1=y^2$ with $y_0>3$, then there exists integers $0< x_1 < y_1$ with $y_1<\sqrt{\frac{y_0+1}{2}}+\frac{1}{2}$ that is also a solution of $2x^2+1=y^2$. Moreover, $x_0,y_0$ can be written as polynomials of $x_1,y_1$.
Question: Is this kind of self reduction phenomenon common for diophantine equation?
By "self reduction", I mean a similar result holds. i.e., if $\mathbf{x}$ is a solution with $||\mathbf{x}||_{some norm}>c$, then there eixsts $\mathbf{x}'$ that is also a solution and $||\mathbf{x}'||_{some norm}<||\mathbf{x}||_{some norm}$. I was told that $y^2-nx^2=1$ is called pell equation. But I didn't find in wiki that all these kind of equations admit such reducible property. I'm also curious about whether it holds for some other diophantine equation. Any references are welcome.