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Topic: conjecture on sums of primes
Replies: 13   Last Post: Apr 18, 2013 12:45 AM

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Posts: 4
Registered: 4/17/13
Re: conjecture on sums of primes
Posted: Apr 17, 2013 3:04 PM
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Butch Malahide wrote:
> 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-Tao_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.

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