Orjan Johansen <firstname.lastname@example.org> wrote: : In article <email@example.com>, : goldbach <firstname.lastname@example.org> wrote: : > : >It is not all that obvious. E.g, assuming the conjecture true, take : >any prime p, no matter how large, then there is an even number E : >such that for E=p + q , q prime and p<=q, p will be the least prime : >in all the pairs of primes whose sum is E. : : How do you prove that?
By Dirichlet's theorem on primes in arithmetic progressions, there is a prime q > p that is congruent to -p modulo (p-1)! . Then E = p+q is divisible by every prime less than p, and so it is not the sum of two primes including a prime less than p.
This doesn't assume Goldbach's conjecture, though.