# Examples of finding algebraic solutions to Diophantine equations.

I saw to prove that $a^2+b^2=c^2$ you can produce the algebraic solutions $a = x^2-y^2, b = 2xy, c = x^2+y^2$. To check that works just multiply it out but it produces infinitely many solutions when you set x and y to be integers.

Can you please show me more examples of this, especially if they show how to find the polynomials!

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Read the section "Rational points on conics" in the motivational chapter in the very beautiful book Rational points on elliptic curves by Silverman and Tate and/or google "rational parametrization of the circle". Also, once you have some knowledge of elliptic curves don't miss Franz Lemmermeyer's Conics - a poor man's elliptic curves which gives a beautiful treatment of Pell conics unified with theory of elliptic curves. Previously fragments of this theory were folklore scattered in pieces throughout the literature.

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