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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.

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

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The 'self reduction' property which you have astutely observed has historically been extremely important in the study of Diophantine equations, both for generating solutions in some cases and for proving impossibility in other cases. In fact, Fermat's use of this in his method of infinite descent was said by Lagrange to be "... one of the most fruitful methods in the whole theory of numbers."

For an elementary treatment of this you could refer to Fermat's method of "descent infinie", The Mathematical Gazette, 95, No.533, June 2011.

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