If the prime factorization of two numbers is known, does a method exist to find the prime factorization of their sum with an asymptotic complexity better than the general prime factorization problem?
In other words, given the prime exponent representations $\{a_i\}$ and $\{b_i\}$ of two integers $a$ and $b$, is there a way to compute $\{c_i\}$ where $c = \prod_i p_i^{c_i} = \prod_i p_i^{a_i} + \prod_i p_i^{b_i} = a + b$ that is faster than naively evaluating the RHS and factoring it?
I have a feeling the answer is obviously no, since if there was a faster way than naively evaluating and applying the state-of-the-art method (such as the general number field sieve), you could probably use this trick by finding two easily factor-able integers to improve the time complexity of the general factorization problem, which would be a contradiction.