Question 946599
By the Goldbach conjecture, we don't need to check 4, 6, 8, ..., 18 (otherwise the Goldbach conjecture would have been proven false). So 2 is the only even number less than 20 that cannot be written as a sum of 2 primes.


Any odd integer that is the sum of two primes must be of the form 2+p where p is an odd prime. Hence, for p = 3, 5, 7, 11, 13, 17 we have 5, 7, 9, 13, 15, 19. Therefore, 1, 3, 11, 17 cannot be written as a sum of two primes.


The sum is 1+3+11+17+2 = 34.