Best Approximations by Quadratic Integers

 Published on Saturday, 18th February 2017, 01:00 pm; Solved by 172;
Difficulty rating: 95%

Problem 591

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.