SOLUTION: Find all natural n such that n^3 + 8 is prime.

Question 1033739: Find all natural n such that
n^3 + 8 is prime.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
.
Since is supposed to be prime, then either
(i) n+2 = 1 and
OR
(ii) and .
The first case (i), the 2nd equation is equivalent to , which when equated to zero will only give the real solution n = -1. This also satisfies the first equation n+2=1. But n = -1 is not a natural number, so we get no solutions from this case.
For the second case, we get and . The first equation has only real solution n = -2, while the second equation has only complex solutions, and clearly n = -2 does not satisfy . Thus, there are no solutions arising from the second case.

Therefore we conclude that there are no natural numbers n such that is prime.

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