, 2022

We note that all of the given possible sums are odd, and also
that every prime number is odd with the exception of 2 (which is
even).

When two odd integers are added, their sum is even.

When two even integers are added, their sum is even.

When one even integer and one odd integer are added, their sum is
odd.

Therefore, if the sum of two integers is odd, it must be the sum of an
even integer and an odd integer.

Since the only even prime number is 2, then for an odd integer to be the
sum of two prime numbers, it must be the sum of 2 and another prime
number.

Note that $$19 = 2 + 17 \qquad 21 = 2 + 19 \qquad 23 = 2 + 21 \qquad 25 = 2 + 23 \qquad 27 = 2 + 25$$ Since
17, 19 and 23 are prime numbers and 21 and 25 are not prime numbers,
then 3 of the given integers are the sum of two prime numbers.