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, ··· .