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 $\lbrace a+b \; | \; a\in A, b\in B\rbrace$)**



prime divisor of elements of $A+B$
This was proved by Erdős and Turán when they were undergraduate students, see Theorem III [here](http://www.renyi.hu/~p_erdos/1934-03.pdf). A stronger result was proved by Győry, Stewart, and Tijdeman, see Theorem 1 [here](http://archive.numdam.org/ARCHIVE/CM/CM_1986__59_1/CM_1986__59_1_81_0/CM_1986__59_1_81_0.pdf).


