At 09:59 AM 11/14/98 -0500, you wrote:
>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]>
>
>
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![http://users.ox.ac.uk/~mert0236/maths/goldbach[2..4].gif](http://users.ox.ac.uk/~mert0236/maths/goldbach[2..4].gif){kind=link}