Countably many

There are NN rational numbers RR such that

5×108+R3+5×108R3 \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 NN?

Details and assumptions

You may use the fact that 0.7937<0.53<0.793701 0.7937 < \sqrt[3]{0.5} < 0.793701

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


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