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.
How hard is it to factorize sum of two numbers
complexity theory;factoring
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.