What is known about the equation $x^2+ay^2=b^2$, where $a$ is a fixed square free positive integer and $b$ is a fixed positive integer.

Question

$(b,0)$ and $(-b,0)$ are two trivial solutions. What do we know about the nontrivial solutions of the equation $x^2+ay^2=b^2$.

Answer 1

Let us write it as $$ x^2 + a y^2 - z^2 = 0 $$ Theorem in Mordell, Diophantine Equations. Theorem 4, page 47, all solutions with $$ \gcd(x,y,z) = 1$$ come in a few parametrizations of the Pythagorean Triple type. On average, it gets more complicated as the (binary) class number gets large. I always do a reality check on a prospective list.

For example, $x^2+ 14 y^2 = z^2.$ We have, at least, one recipe with $z = u^2 + 14 v^2,$ you can probably work out the recipes for $x$ and $y.$ On the other hand, we can have $z = 2 u^2 + 7 v^2$ and new $x,y$

Gets new types when $h(-4a) $ is divisible by $3$. We could have $z = u^2 + 11 v^2.$ We get something different by taking $z = 3 u^2 + 2uv + 4 v^2$ Let me check some things...

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On average, more such recipes are needed