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Fix a positive integer $k\geq 2$. Can we always find infinitely many primes $p_1,p_2,\ldots$ such that the sum of any $k$ of them is composite?

If $k$ is even, it is clear that we can. We only need to not choose $2$, so that all the chosen primes are odd, and the sum of any $k$ of them is even, hence composite. But what about if $k$ is odd?

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Choose the primes so that they are all congruent to $1$ modulo $k$. That we can find infinitely many is a consequence of Dirichlet's Theorem on primes in arithmetic progressions.

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