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Problem 591

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Best Approximations by Quadratic Integers

Given a non-square integer $d$, any real $x$ can be approximated arbitrarily close by quadratic integers $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate $\pi$ with precision $10^{-13}$:
$4375636191520\sqrt{2}-6188084046055 < \pi < 721133315582\sqrt{2}-1019836515172 $
We call $BQA_d(x,n)$ the quadratic integer closest to $x$ with the absolute values of $a,b$ not exceeding $n$.
We also define the integral part of a quadratic integer as $I_d(a+b\sqrt{d}) = a$.

You are given that:

• $BQA_2(\pi,10) = 6 - 2\sqrt{2}$
• $BQA_5(\pi,100)=26\sqrt{5}-55$
• $BQA_7(\pi,10^6)=560323 - 211781\sqrt{7}$
• $I_2(BQA_2(\pi,10^{13}))=-6188084046055$

Find the sum of $|I_d(BQA_d(\pi,10^{13}))|$ for all non-square positive integers less than 100.

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二次整数最佳逼近

给定一个非完全平方的整数$d$，任意实数$x$可以用所谓二次整数$a+b\sqrt{d}$任意逼近，其中$a,b$均为整数。例如如下不等式能够以$10^{-13}$的精度逼近$\pi$：
$4375636191520\sqrt{2}-6188084046055 < \pi < 721133315582\sqrt{2}-1019836515172$
我们记$a,b$不超过$n$的所有二次整数中最接近$x$的为$BQA_d(x,n)$。
同时我们定义二次整数的“整数”部分为$I_d(a+b\sqrt{d}) = a$。

已知：

• $BQA_2(\pi,10) = 6 - 2\sqrt{2}$
• $BQA_5(\pi,100)=26\sqrt{5}-55$
• $BQA_7(\pi,10^6)=560323 - 211781\sqrt{7}$
• $I_2(BQA_2(\pi,10^{13}))=-6188084046055$

对于所有小于100的非完全平方整数$d$，求其对应的$|I_d(BQA_d(\pi,10^{13}))|$之和。

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