SOLUTION: What is the sum of all positive integers less than 20 that cannot be written as the sum of 2 prime numbers?

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Question 946599: What is the sum of all positive integers less than 20 that cannot be written as the sum of 2 prime numbers?
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
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.