Binary Quadratic Form II

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 $C(10^{14})$.

The task involves pretty advanced mathematics, especially number theory. Answering it requires a good understanding of algebraic forms and their representations. It seems like it involves the theory of quadratic forms and modular forms.

The binary quadratic form is $f(x,y)=x^2+5xy+3y^2$ and a primitive representation is defined as involving coprime numbers $x$ and $y$.

We need to tally the number of primitive representation of all squares not exceeding a given number. Given values are $C(10^3)=142$ and $C(10^{6})=142463$, and we are asked to find $C(10^{14})$.

Unfortunately, the computation of $C(10^{14})$ is not quite straightforward nor is it easy. It involves complex analyses and computational methods which are beyond the standard curriculum and also is quite difficult to perform by hand. You’d typically need a symbolic computation system set up for advanced mathematics computations to get this done, like SageMath, Mathematica or something similar.

You should note that the task is not of the ordinary homework/calculus type and should be approached appropriately. I’d recommend finding someone who has a strong background in number theory, as this problem involves the study of numbers and the properties of numerical relationships.

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