If we know the prime factorizations of $a$ and $b$, can we say anything about the prime factorization of $a + b$? I know we can say if $a$ and $b$ have a common factor, then that factor is also in $a+b$. But my guess is we can't say much more than this. Is it even possible that the factors of $a$ and $b$ completely determine the factors of $a+b$? Would finding such a relationship solve most open questions and be equivalent to finding a formula for finding primes?
Sorry if the last part is too opinion basd, i can edit it down if needed.
 A: Whenever $a$ and $b$ are relatively prime, the factors of $a+b$ can be anything.
For instance, in this case there will be infinitely many primes of the form $a+nb$ (Dirichlet's theorem), so there's no way this factorization can be determined.
So, I think in general the answer to your question is no (it's true of course for trivial cases like $a = nb$).
A: I think you can say a little bit.  We at least know that any prime-power that divides both $a$ and $b$ will divide $a+b$.  And we know that any prime-power that divides exactly one of $a$ and $b$ can not divide $a+b$.  Given factorizations of $a$ and $b$, we know some prime-powers that definitely are and some that definitely are not in the factorization of $a+b$.
In fact, this is the idea behind Mersenne and Fermat primes.  And the search for primes of the form $n! \pm 1$.  We grab a number with lots of small prime factors and add one, knowing that the result won't have any of those small prime factors and so we have a better chance of getting a prime.
A: The abc conjecture states

Let $\varepsilon>0$ and $(a,b,c)$ be a triple of coprime positive integers such that $a+b=c$. Calling $P$ the product of the distinct prime factors of $abc$, we have $c>P^{1+\varepsilon}$ only for finitely many $(a,b,c).$

In other words, it appears that usually $c$ has relatively small (or few) prime factors. Shinichi Mochizuki developed a brand new theory within four long papers on which he claims to have proved the conjecture. Recently, a handful of mathematicians have announced they are going through his work.
A: The abc conjecture says that if $a$ and $b$ are composed of prime factors with large multiplicity, then $a+b$ can't also be composed of prime factors with large multiplicity.
