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1. Given $M$ is it possible to pick a set of $T>M$ distinct numbers $a_i\in\Bbb Z$ such that sum of any $M+1$ or more of them will always be a prime and sum of any $M$ or less of them is always composite with no prime factors bigger that $M^\frac{1}c$ with some fixed $c>1$?

2. Given $M$ is it possible to pick a set of $T>M$ distinct numbers $a_i\in\Bbb Z$ such that sum of any $M$ or less of them will always be a prime and sum of any $M+1$ or more of them is always composite with no prime factors bigger that $M^\frac{1}c$ with some fixed $c>1$?

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Let $M=8$. We show there is no set $T$ of $9$ positive integers such that the sum of any $8$ is prime. If $T$ has only odd integers, we have a problem, and if $T$ has only even integers, we have a bigger problem.

So there is at least one even integer $e$ and at least one odd integer $o$. Consider the set $T\setminus{e}$. The sum of its elements is odd. But then the sum of the elements of $T\setminus {o}$ is even.

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  • $\begingroup$ What if we ask complement question? $\endgroup$ – Turbo Feb 12 '15 at 6:19
  • $\begingroup$ I do not know the exact statement of the complement question you are contemplating. $\endgroup$ – André Nicolas Feb 12 '15 at 6:21
  • $\begingroup$ I see you have formulated the complement question. Again pick $M$ almost anything,say $8$ or $9$. The sum of $M$ or fewer cannot be always prime. If there are $2$ or more even, their sum is no prime. And if there are $2$ odd their sum is not prime. $\endgroup$ – André Nicolas Feb 12 '15 at 7:01

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