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There are NNN rational numbers RR R such that
5×108+R3+5×108−R3 \sqrt[3] {5\times 10^8 + \sqrt{R} } + \sqrt[3]{5\times 10^8- \sqrt{R} } 35×108+R+35×108−R
is an integer. What are the last three digits of NNN?
Details and assumptions
You may use the fact that 0.7937<0.53<0.793701 0.7937 < \sqrt[3]{0.5} < 0.793701 0.7937<30.5<0.793701
We choose the real value of the roots (where applicable).
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