# Can you find the factors of the sum of 2 integers given their factors?

Given the factors of $2$ integers $a$ and $b$, can you find the factors of the sum of $a$ and $b$ without first multiplying the factors to get $a$ and $b$? This is just something I recently thought of and became curious about. If it is possible, how would one go about doing it?

Example: factors of $a: 3,7$ and factors of $b : 2,2,3$. What are factors of $a + b$ without first doing $3 \cdot 7$ and $2 \cdot 2 \cdot 3$ and then adding them together and then factoring it?

EDIT: I slightly mis-worded my question and I think I fixed the wording.

• $6$ isn't a prime factor. You can reduce your work by finding common factors ($3$ in your case), but from there it's... tricky. When $a+b=c$, statements about factors of $a$, $b$ and $c$ are usually something related to abc conjecture. – Abstraction Dec 4 '15 at 14:38
• @Abstraction Woops, that 6 was supposed to be a 3. I fixed it. – EnderShadow Dec 4 '15 at 15:14
• 3*(7+2*2). Don't know that you'll find an easier way. – Morgan Rodgers Dec 4 '15 at 15:17
• @MorganRodgers I realized I slightly misworded the question. – EnderShadow Dec 4 '15 at 17:04

You won't find an easier way. If you could, then you'd have an efficient factorisation algorithm, since any odd integer $n$ is $\frac{n+1}{2} + \frac{n-1}{2}$; inductively, we'd be able to factor $\frac{n \pm 1}{2}$ efficiently, and hence $n$ efficiently. But integer factorisation is a notoriously hard problem (on which the security of the Internet still rests, in many places!), so if there were an easy way like this, we'd have heard about it already.