Suppose you had the prime factorization of $x = \prod_{k=1}^m p_k$. Are there any algorithms for deciding the prime factorization of $x + 1$? Are their prime factorizations related in any way at all?
EDIT: It's known that $x$ and $x + 1$ are coprime. That means that you get some amount of information about the prime factors of $x + 1$. Can the information about prime factors of $x$ be used to figure out the prime factors of $x + 1$ faster than just "prime sieve" + "filter out the factors of $x$"?