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

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gnasher729

Posts: 417
Registered: 10/7/06
Re: conjecture on sums of primes
Posted: Feb 3, 2012 7:06 PM
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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.



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