A number theory problem by Kenneth Gravamen

The number asked can be expressed as $x^{3} - 1 = (x - 1)(x^{2} + x + 1)$ . Given that it is prime, either $x - 1$ or $x^{2} - x + 1$ must be 1 . Solving for x $x - 1 = 1 \rightarrow x = 2$ $and$ $x^{2} + x + 1 = 1 \rightarrow x = -1, 0$ By substituting the first value of x results $x^{3} - 1 = 2^{3} - 1 \rightarrow 7$ . Second value and third value, $(-1)^{3} - 1 = -2$ and $(0)^{3} - 1 = -1$ . Two of this values are not accepted because they are negative. Hence, our answer is $\boxed{7}$ .