Problem

Source: Iranian third round midterm number theory exam problem 3

Tags: number theory, kobayashi



Let $S$ be an infinite set of positive integers and define: $T=\{ x+y|x,y \in S , x \neq y \} $ Suppose that there are only finite primes $p$ so that: 1.$p \equiv 1 \pmod 4$ 2.There exists a positive integer $s$ so that $p|s,s \in T$. Prove that there are infinity many primes that divide at least one term of $S$.