Countably many

Number Theory Level 5

There are $N$ rational numbers $R$ such that

$\sqrt[3] {5\times 10^8 + \sqrt{R} } + \sqrt[3]{5\times 10^8- \sqrt{R} }$

is an integer. What are the last three digits of $N$ ?

Details and assumptions

You may use the fact that $0.7937 < \sqrt[3]{0.5} < 0.793701$

We choose the real value of the roots (where applicable).