Re: Mersenne: Goldbach's Conjecture

[Fred W. Helenius]

At 11:16 AM 11/14/98 -0000, Thomas Womack wrote:
[snip]
>I've ran a test for N<2^26, which took a night on a P2/233, and produced the
>data graphed in http://users.ox.ac.uk/~mert0236/maths/goldbach.gif and
>http://users.ox.ac.uk/~mert0236/maths/goldbach[2..4].gif. What I'm plotting
>there is the probability density function of (R(x) log(x)^2)/x; you will
>notice the pretty fractal structure, which I am at the moment utterly unable
>to explain.

This structure is caused by the presence/absence of small prime factors
in n.  For example, if n is a multiple of three, there are three ways
that two numbers considered modulo 3 can add up to n; their residues
can be 0 and 0, 1 and 2, or 2 and 1.  The first case is excluded, since
primes (with one exception) aren't multiples of three; so we have two
cases where the addends could be prime.  If n == 1 (mod 3), the
possibilities are 0 and 1, 1 and 0, or 2 and 2.  Here, only in the last
case could the addends be prime; and similarly for n == 2 (mod 3).
So, other things being equal, one expects an even multiple of three to
be expressible as a sum of two primes twice as often as an even
non-multiple of three.  I think you'll find that this produces the first
split in your graphs; roughly, between the values above 3 and those below.

For other small primes p, one finds that being a multiple produces a
factor of (p-1)/(p-2) advantage over a non-multiple.  So a number
divisible by many small primes can expect many more ways of being
expressed as the sum of two primes.  My guess is that the high point on
your first graph near 30000 is actually 30030 = 2*3*5*7*11*13.  It looks
as if most of the points near y = 6 are multiples of 2310 = 2*3*5*7*11;
you can see the regular spacing.

--
Fred W. Helenius        <[EMAIL PROTECTED]>