Is $83^{27} +1 $ a prime number?
Solution 1:
$83$ is odd, so is any power of $83$. Hence $83^{27}+1$ is even, but the only even prime number is $2$ and this number is not $2$.
More generally, if $a,k\in\mathbb N$ and $k$ is odd, then $$a^k+1\equiv (-1)^k+1\equiv 0\pmod{a+1}$$ So $a+1\mid a^k+1$. In this case this yields $84=2^2\cdot 3\cdot 7$ as divisor.
Solution 2:
Well, it is an even number, so...