Question 115
Let $S$ be an infinite subset of natural numbers. Define the set $S'$ as follows:
$$
S' = \{x^y + y^x \mid x, y \in S, x \ne y\}.
$$
Prove that there are infinitely many prime numbers $p$ such that $p$ divides at least one element of $S'$.
$$
S' = \{x^y + y^x \mid x, y \in S, x \ne y\}.
$$
Prove that there are infinitely many prime numbers $p$ such that $p$ divides at least one element of $S'$.