Consider the following binary quadratic form:
$$ \begin{align} f(x,y)=x^2+5xy+3y^2 \end{align} $$A positive integer $q$ has a primitive representation if there exist positive integers $x$ and $y$ such that $q = f(x,y)$ and $\gcd(x,y)=1$.
We are interested in primitive representations of perfect squares. For example:
$17^2=f(1,9)$
$87^2=f(13,40) = f(46,19)$
Define $C(N)$ as the total number of primitive representations of $z^2$ for $0 < z \leq N$.
Multiple representations are counted separately, so for example $z=87$ is counted twice.
You are given $C(10^3)=142$ and $C(10^{6})=142463$.
Find $C(10^{14})$.
Solution
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