Problem 769

Binary Quadratic Form II

Consider the following binary quadratic form:

$$f(x,y)=x^2+5xy+3y^2$$

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 $C(10^{14})$.

二元二次型II

考虑如下二元二次型：

$$f(x,y)=x^2+5xy+3y^2$$

对于正整数$q$，若存在正整数$x$和$y$使得$q=f(x,y)$且$\gcd(x,y)=1$，则称之为$q$的本原表达式。

我们感兴趣的是完全平方数的本原表达式。例如：

$17^2=f(1,9)$

$87^2=f(13,40) = f(46,19)$

记$C(N)$为所有满足$0 < z \leq N$的完全平方数$z^2$的本原表达式数目之和。

同一个数的不同本原表达式单独计入，因此如$z=87$就有两组本原表达式。

已知$C(10^3)=142$和$C(10^{6})=142463$。

求$C(10^{14})$。