How hard is it to factorize sum of two numbers Say I have numbers with known factorizations $n = \prod \limits _i p_i ^{n_i}$ and  $m = \prod \limits _i p_i ^{m_i}$ (where $p_i$ is the $i$th prime).

How hard is it to factorize $m+n$? Is there a more intelligent algorithm than if factorizations of $m$ and $n$ were not known? Assume $n$ and $m$ coprime as it is trivial to make them so.

The fact that $m+n$ will share no factors with $n$ or $m$ seems very helpful for small numbers, but I doubt it offers much for large ones.
There is currently no known (asymptotical) more intelligent algorithm (and it is also not expected that there should be one) than if factorizations of $m$ and $n$ were not known (assuming $m$ and $n$ to be coprime). Even the case where $2$ is a prime factor doesn't count, because checking some of the smallest prime numbers can be done without much effort anyway.