Hunting for Primeary Sets

I call a set P of positive integers primeary if the sum of an odd number of elements of P is always prime.

It's clear what the sum of an odd number k > 1 of integers means; for k = 1, the sum of an integer can be taken as the integer itself. Hence, each element of a primeary set must be a prime number. Obviously, the sum of an even number of primes distinct from 2 is always even, so only sums of an odd number of terms are considered in the definition of primeary set.

The order of a primeary set is the number of elements in the set.

Of course, the fundamental problem on primeary sets is:

For each positive integer n, is there a primeary set of order n?

Here are examples of primeary sets up to order 4.

order 1: {2}

order 2: {2, 3}

order 3: {3, 5, 11}

order 4: {5, 7, 11, 181}

Can you extend this list?

Jo. Pe

No.

Proof: If the set contains three primes which are all equal modulo 3, then their sum will be divisible by 3. If the set contains three primes which are all different modulo 3, then their sum will be divisible by 3. So the largest set we can construct has primes which modulo 3 are (1,1,2,2).