Show that no integer of the form $a^3 +1$ is a prime for $a>1$

Question

Can someone please solve this and explain the steps taken to reach this solution ?

Answer 1

Factorise $$a^3 +1 \equiv (a+1)(a^2 -a +1)$$

This is made up of the product of two integers if $a > 1$, namely $a+1$ and $a^2 -a + 1$. i.e: it is the very definition of a composite number. (that is, unless $a+1$ or $a^2 - a + 1$ is equal to $1$, but the hypothesis $a>1$ immediately gives you that.

In fact, it's normally a good idea to factorise $$a^n + b^n = (a+b)(a^{n-1} - a^{n-2}b - \cdots - ab^{n-2} + b^{n-1})$$

in these sort of situations.

Answer 2

Hint: $a^3+1=(a+1)(a^2-a+1)$, so...