## High powers of irrational numbers

### Problem 721

Given is the function $f(a,n)=\lfloor{(\lceil{\sqrt{a}\:\rceil}+\sqrt{a}\:)^n}\rfloor$.
$\lfloor{.}\rfloor$ denotes the floor function and $\lceil{.}\rceil$ denotes the ceiling function.
$f(5,2)=27$ and $f(5,5)=3935$.

$G(n) = \displaystyle \sum_{a=1}^n f(a, a^2).$
$G(1000) \text{ mod } 999\,999\,937=163861845.$
Find $G(5\,000\,000).$ Give your answer modulo $999\,999\,937$.