
Re: conjecture on sums of primes
Posted:
Apr 17, 2013 3:15 PM


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 spacedout evenly, for any size K. How can you say that there are more than N ways to buildup 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 sortof 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 nextoneup to the nextonedown, and you get the same integer E! And so on, provided you have enough (K) primes to make at least N+1 pairs. And GreenTao Theorem provides us that there's *at least one* sequence of K primes for any K arbitrarilylarge. K >= (N+1)*2, no problem.

