Folklore

It is known that $b=2013^{2013}+2$. Will the numbers $b^{3}+1$ and $b^{2}+2$ be coprime?
The number 2013 is divisible by 3, since the sum of its digits equals 6. Therefore, $b^{3} \equiv-1(\bmod 3)$. This means, $b^{3}+1 \equiv-1+1=0$ $(\bmod 3)$ and $b^{2}+2 \equiv 1+2=0(\bmod 3)$.

Thus, these numbers have a common divisor of 3.

## Answer

They will not.