Taking the Cramer model, the probability that a number N is not expressible as the sum of two primes is product(x=1...N-1) (1 - 1/log(x)log(N-x)) < product(x

Taking the Cramer model, the probability that a number N is not expressible as the sum of two primes is product(x=1...N-1) (1 - 1/log(x)log(N-x)) < product(x=1...N-1) (1 - 1/log(N)) = (1 - 1/log(N)) ^ (N-1) Which is eventually below (1 - 1/log(N)) ^ log(N) ^ sqrt(N) Which converges to (1/e) ^ sqrt(N) And so is eventually below (1/2) ^ sqrt(N) The -log of the probability that a number is expressible as the sum of two Cramer numbers is thus at most -log(1 - (1/2) ^ sqrt(N)) Which is eventually below 2 * (1/2) ^ sqrt(N) You can show that this is a convergent series, and if the -log of the probability converges then the probability that ALL integers (minus perhaps the first few) have this property must converge, to something above zero. Thus, there is a nonzero chance that a set of Cramer numbers will have the property that all a finite number of positive integers will be expressible as the sum of two of them. Thus, prime numbers do not need any magical properties for some version of the Goldbach conjecture to be correct on them - the magical properties are only necessary to actually prove it.