# Are There Infinitely Many Primes of a Certain Form?

Definition. Let $S_n = 3^n - 2^n$ for every positive integer $n$.

Question. Are there infinitely many primes $p$ such that $S_p$ is (not) prime?

Some Facts.
(a) For any positive integer $n$ such that $S_n$ is prime, $n$ must clearly be prime.
(b) There are primes $p$ for which $S_p$ is (not) prime.

Generalization of the Original Question

Lemma. Let $x$, $y$, $n$ be integers such that $x > y > 0$ and $n > 1$. If $x^n - y^n$ is prime, then both $n$ is prime and $x - y = 1$.

Conjecture. Let $x$ be a positive integer. Then there are infinitely many primes $p$ such that $(x+1)^p - x^p$ is (not) prime.

An interesting question would be which conjectures hold for $\frac{x^n - y^n}{x - y}$.

• I would like to believe that this is as hard or harder than the question "Are there infinitely many Mersenne primes?", which is an unsolved problem. – user17762 Jan 14 '13 at 19:18
• There's a few relevant references at oeis.org/A057468. The answer doesn't seem to be known. – Chris Eagle Jan 14 '13 at 19:26