The following result is proved: If
![$A\subseteq \{
1,\, 2,\, \ldots ,\, n\} $](abs/img1.gif)
is the subset of largest cardinality such
that the sum of no two (distinct) elements of
![$A$](abs/img2.gif)
is prime, then
![$\vert A\vert=\lfloor(n+1)/2\rfloor$](abs/img3.gif)
and all the elements of
![$A$](abs/img2.gif)
have the
same parity. The following open question is posed: what is the
largest cardinality of
![$A\subseteq \{
1,\, 2,\, \ldots ,\, n\} $](abs/img1.gif)
such that the sum of no two (distinct) elements of
![$A$](abs/img2.gif)
is prime
and
![$A$](abs/img2.gif)
contains elements of both parities?
Received April 16 2008;
revised version received December 13 2008.
Published in Journal of Integer Sequences, December 13 2008.