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Problem 804

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Counting Binary Quadratic Representations

Let $g(n)$ denote the number of ways a positive integer $n$ can be represented in the form:
$$x^2+xy+41y^2$$
where $x$ and $y$ are integers. For example, $g(53)=4$ due to $(x,y)\in (-4,1),(-3,-1),(3,1),(4,-1)$.

Define ${\displaystyle T(N)=\sum _{n=1}^{N}g(n)}$. You are given $T({10}^3)=474$ and $T({10}^6)=492128$.

Find $T({10}^{16})$.

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二元二次型表示计数

记$g(n)$为将正整数$n$表示成如下二元二次型的方式数目：
$$x^2+xy+41y^2$$
其中$x$和$y$均为整数。例如，$g(53)=4$，对应的方式为$(x,y)\in (-4,1),(-3,-1),(3,1),(4,-1)$。

定义${\displaystyle T(N)=\sum _{n=1}^{N}g(n)}$。已知$T({10}^3)=474$，$T({10}^6)=492128$。

求$T({10}^{16})$。

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