# Problem 591: 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.