4. How many prime numbers can be expressed in the form $n^{n+1}+1$, with $n$ a positive integer?
(A) 0
(B) 1
(C) 2
(D) more than 2, but finite in number
(E) infinite.
4. The answer is (B). Consider the case where $n$ is odd. Then any of its powers is odd, and the next number is even. The only even prime number is 2, and it can only be obtained in the case $n=1$. Now consider the case where $n$ is even. In this case, $n+1$ is odd, and it is thus possible to perform the factorization $n^{n+1}+1=(n+1) \cdot Q(n)$. If $n$ is an even number different from 0, $n+1$ is a number greater than 2 and $Q(n)$ is a number greater than 1, so $n^{n+1}+1$ is not a prime number.

(Busoni)