conjecture on sums of primes

[Paul]

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?

Is it equivalent to a better-known conjecture? Has exactly the same
conjecture already been made and given a standard name?

Thank you,

Paul Epstein

[Richard Tobin]

In article <04113ffd-b6f8-4092...@z31g2000vbt.googlegroups.com>,

Paul <peps...@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.

To convince yourself (rather than prove) that this is true, consider

http://www.cogsci.ed.ac.uk/~richard/goldbach.html

Since creating that graph, I have learnt that it is known as Goldbach's
Comet.

-- Richard

[Paul]

On Feb 2, 10:46 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <04113ffd-b6f8-4092-a939-55aff8c18...@z31g2000vbt.googlegroups.com>,

>
> 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.
>
> To convince yourself (rather than prove) that this is true, consider
>
>  http://www.cogsci.ed.ac.uk/~richard/goldbach.html
>
> Since creating that graph, I have learnt that it is known as Goldbach's
> Comet.

Your graph is interesting, but it may be possible to show that
Goldbach's Comet can mislead the intuition.
For example, if there is a very large x (say > 10^18) which can be
expressed as the sum of two primes in very few ways (for example < 10
ways), then the limitations of such graphs would be shown.

Does anyone know of any large near counter-examples to Goldbach's
conjecture? For example, what's the largest known even number that
can be expressed as the sum of two primes in only 1 way?

Does anyone know any results/tables for the largest known even number
E_k that can be expressed as the sum of two primes in <= k ways?

Paul Epstein

[Richard Tobin]

In article <8faccb56-a966-4767...@o14g2000vbo.googlegroups.com>,

Paul <peps...@gmail.com> wrote:

>Your graph is interesting, but it may be possible to show that
>Goldbach's Comet can mislead the intuition.
>For example, if there is a very large x (say > 10^18) which can be
>expressed as the sum of two primes in very few ways (for example < 10
>ways), then the limitations of such graphs would be shown.

Though Goldbach's conjecture has been verified up to about that number,
I don't of anyone hving calculated the number of ways anything like that
far.

>Does anyone know of any large near counter-examples to Goldbach's
>conjecture?

I have not heard that there are any.

>For example, what's the largest known even number that
>can be expressed as the sum of two primes in only 1 way?

As far as I know, 12.

-- Richard

[Don Redmond]

I seem to recall some results (from the 60s or 70s) to the effect that
on average the number of representations is unbounded, that is, the
larger
the number the more representations. Unfortunately I can't give you
any
exact references.

Don

[Thomas Nordhaus]

Cf. OEIS A000954 for example. I made a little survey, looking for the
"largest relative drop in the number of Goldbach-partitions r(2n)
between consecutive even numbers". Maybe that counts as
"near-counterexample".

In the range 4 <= 2n <= 10000 I found r(630)=41, r(632)=10. Actually
n=632 is conjectured to be the largest n with exactly 10 partitions, see
A000954.

--
Thomas Nordhaus

[christian.bau]

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.

[Butch Malahide]

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

[Richard Tobin]

In article <9e8afa51-f34f-4fab...@g27g2000yqa.googlegroups.com>,

Butch Malahide <fred....@gmail.com> wrote:

>It follows from the existence of arbitrarily long arithmetic
>progressions of primes:

That's neat.

-- Richard

[pima...@gmail.com]

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.

[david petry]

On Thursday, February 2, 2012 1:35:43 AM UTC-8, Paul 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.

Here's a stronger conjecture which a probabilistic argument suggests is true:

For every N, there is an M, such that every even integer greater than M can be expressed as a sum of two primes in at least N different ways.