On the Natural Numbers that cannot be Expressed as a Sum of Two Primes

In this paper, a structural property of prime numbers will be introduced and analyzed. Our analysis is based on the application of the sequence of even integers 2,10,16,22,26,28 which is defined as follows: F(0) = 2, for i 0, F(i) = F(i 1)+ l, where l is the smallest even integer that is valid for: F(i 1)+ 2k 1 P, for k = 1, 2, . . . l/2-1, F(i 1)+ 2k 1 P, for k = 2l and P is the set of prime numbers. The sequence F answers the question, which are the natural numbers that cannot be expressed as a sum of two primes. We prove that numbers expressed as j = F(i)+ 1 cannot be written as the sum of two primes, for i = 1, 2, .