# Must a decision problem in \$NP\$ have a complement in \$Co-NP\$, if I can verify the solutions to in polynomial-time?

Goldbach’s Conjecture says every even integer $$>$$ $$2$$ can be expressed as the sum of two primes.

Let’s say $$N$$ is our input and its $$10$$. Which is an integer > 2 and is not odd.

## Algorithm

1.Create list of numbers from $$1,to~N$$

2.Use prime-testing algorithm for creating a second list of prime numbers

3.Use my 2_sum solver that allows you to use primes twice that sum up to $$N$$

``for j in range(list-of-primes)):   if N-(list-of-primes[j]) in list-of-primes:    print('yes')    break ``

4.Verify solution efficently

``if AKS-primality(N-(list-of-primes[j])):     if AKS-primality(list-of-primes[j]):         print('Solution is correct') ``

5.Output

``yes 7 + 3 Solution is correct ``

## Question

If the conjecture is true, then the answer will always be Yes. Does that mean it can’t be in $$Co-NP$$ because the answer is always Yes?