Finding the sum of two numbers knowing only the primes

Question

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.

Answer 1

You can make it a little easier by looking for common factors. In your example, $2$ is an element of both factorizations, so will be a factor of the sum. If there are lots of common factors, that will help a lot. Otherwise, not so much ...

Answer 2

This might get you closer:

$$\frac{1}{30}+\frac{1}{8}=\frac{(30 + 8)}{gcd(30 +8,lcm(30,8))}=\frac{19}{lcm(30,8)}=\frac{19}{120}$$

where $gcd(\cdot,\cdot)$ is greatest common denominator and $lcm(\cdot,\cdot)$ is the lowest common multiple

Answer 3

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.