On the one hand, the Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. On the other hand, P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? In computational complexity theory, another major complexity class is ASPACE(S(n)) for some S(n). It is known that ASPACE(log n) = P. We prove if the complexity class NP is equal to ASPACE(S(n)) for some S(n) = o(log n), then the weak Goldbach's conjecture is false. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then we obtain that NP is not equal to ASPACE(S(n)) for all S(n) = o(log n).