Given $S$ of positive integers $>$ $1$ is there some combination with even $SUM$ > $2$ that is NOT the sum of two primes?
$SUM$ = 10
$S$ = $[4,6]$
Sum of Two Primes $5 + 5 = 10$.
It has an $O(1)$ algorithm if Goldbach is True (always output $NO$). Otherwise, it would seem to be NP-complete because it would require solving Subset-Sum.
Will a many-one-reduction from $Subset-sum$ work for this decision problem in poly-time?