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Problem 769

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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 $0<z\le N$.

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 $C({10}^{14})$.

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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$0<z\le N$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$C({10}^{14})$。

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