Consider two infinite subsets $A$ and $B$ of natural numbers. I am positive that the set of prime divisors of elements of $A+B$ is infinite but I can not prove it. Maybe this is a known result. If anybody has an idea or a reference please let me know. (by $A+B$ I mean $\lbrace a+b \;  \; a\in A, b\in B\rbrace$)
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
This was proved by Erdős and Turán when they were undergraduate students, see Theorem III here. A stronger result was proved by Győry, Stewart, and Tijdeman, see Theorem 1 here. 

