Summary: The following result is proved: If $A\subseteq \{ 1,\, 2,\, \ldots ,\, n\} $ is the subset of largest cardinality such that the sum of no two (distinct) elements of $A$ is prime, then $\vert A\vert=\lfloor(n+1)/2\rfloor$ and all the elements of $A$ have the same parity. The following open question is posed: what is the largest cardinality of $A\subseteq \{ 1,\, 2,\, \ldots ,\, n\} $ such that the sum of no two (distinct) elements of $A$ is prime and $A$ contains elements of both parities?

primes, sumsets, distribution of primes