Are there any theorems that give us any information about the prime factorization of some integer $n+1$, if we already know the factorization of $n$?
Recalling Euclid's famous proof for the infinity of the set of prime numbers, I guess we know that if $n = p_1 p_2 p_3$, then $n+1$ cannot have $p_1$, $p_2$, or $p_3$ as factors. But is there any way we could use the information about $n$'s factorization to determine something more precise about the factorization of $n+1$?