I'm looking for a formula to generate all solutions $x$, $y$, $z$ for $x^2 + y^2 = 5z^2$.
Any advice?
abstract-algebradiophantine-equations
asked 2012-10-31
user id:47636
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Consider the easier question of finding all integer solutions to $x^2 + y^2 = z^2$. There are two standard ways of doing this. One way is to rewrite it as $x^2 = (z + y)(z - y)$ and compare the prime factorizations of both sides. But that method doesn't work so well for the present equation, where when you try to do this you get $x^2 = (\sqrt{5} \cdot z - y)(\sqrt{5} \cdot z - y)$ and have to work in a number system other than the integers... – 2012-10-31
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The other standard way to find all integer solutions to $x^2 + y^2 = z^2$ is to recast the problem as that of finding all *rational* solutions to $X^2 + Y^2 = 1$. See, e.g. http://www.ux1.eiu.edu/~cfdmb/4900/rational.pdf (but note that this document omits the geometric picture). This method is more promising for finding all integer solutions to the equation that you wrote down above. Just curious - in what context did this question come up? – 2012-10-31