# Self reducible diophantine equation

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.

• in a well order these properties would lead to contradiction.if $y_0$ was assumed least.
– user645636
Dec 8, 2019 at 12:50
• recommend Weissman bookstore.ams.org/mbk-105 Dec 8, 2019 at 14:07