TITLE: Finding the sum of two numbers knowing only the primes
QUESTION [6 upvotes]: Pretend $N_1$ is the prime factorization of 30 and $N_2$ is the prime factorization of 8. Is there a way, using only $N_1$ and $N_2$, to get the prime factorization of the sum, 38?
It is easy to do product (just merge the prime factors) but I do not know about addition.

REPLY [2 votes]: What you want is pretty much hopeless. If $N_1$ and $N_2$ both have easy but disjoint factorisations (many small factors but not the same ones in $N_1$ and $N_2$), then $N_3$ is quite likely to involve by contrast some very large prime factor(s). The ABC-conjecture makes a precise (but not so easy to formulate) statement to this effect. It hasn't been proved, but then it hasn't been disproved either.