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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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up vote 12 down vote accepted

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.

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Thank You and May God Bless Erdos. –  user37755 Jul 26 '13 at 13:27

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