On Feb 4, 2012, Butch Malahide wrote: > On Feb 2, 3:35 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? > > It follows from the existence of arbitrarily long arithmetic > progressions of primes: > > https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem
Yes! I found it as I was *describing* the problem to ask for help from this thread :) :
There's a sequence of K primes, all spaced-out evenly, for any size K. How can you say that there are more than N ways to build-up the same integer using those primes taken 2 at a time? (for any N). (You must use only *those* primes because an 'outside' prime plus one of those primes, would produce a bunch of different integers, over all those primes).
The answer is sort-of a Gaussian thing: You can take almost any two of the K primes, let's say the bottom and the top ones. Those two added together make an integer E. Add the next-one-up to the next-one-down, and you get the same integer E! And so on, provided you have enough (K) primes to make at least N+1 pairs. And Green-Tao Theorem provides us that there's *at least one* sequence of K primes for any K arbitrarily-large. K >= (N+1)*2, no problem.