The Complexity of Goldbach's Conjecture

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. A major complexity classes are P and NSPACE(S(n)) for some S(n). We prove if the complexity class P is equal to NSPACE(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 P is not equal to NSPACE(S(n)) for all S(n) = o(log n).