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