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 $19 = 2 + 17$, $21 = 2 + 19$, $23 = 2 + 21$, $25 = 2 + 23$, $27 = 2 + 25$. 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.