On Feb 2, 9:35 am, Paul <pepste...@gmail.com> wrote: > I conjecture that, for all integers N > 1, there exists an integer E > such that E can be expressed as the sum of two primes in more than N > different ways. > > Is this conjecture true, false, or unknown?
There are about n / ln (n) primes p <= n. Therefore there about (n^2 / 2 ln^2 (n)) sums of two primes p <= q <= n. Therefore there more than about (n^2 / 2 ln^2 (n)) sums of two primes p + q <= 2n, with p <= q; almost all the sums are even. Therefore there are even numbers <= 2n that can be expressed as the sum of two primes in about (n / 2 ln^2 (n)) ways.
Given n, solve N = n / 2 ln^2 (n)), giving n about 2N ln^2 (N) (roughly), so there should be an E <= 4N ln^2 (N).
I'm sure you can make this rigourous and a bit more precise, but the conjecture is definitely true.