What are the prime numbers $p$ such that $p$ divides $29^{p}+1$?
Let $p$ be a prime number, then by Fermat's little theorem $29^{p} \equiv 29 \pmod{p}$, so $p \mid 29^{p}+1$ if and only if $p \mid 29+1=30$. Therefore, the prime numbers $p$ that satisfy the condition are exactly the prime divisors of 30.