[extropy-chat] Goldbach's Conjecture Resolved! A Story

[spike]

> bounces at lists.extropy.org] On Behalf Of Lee Corbin
> Subject: [extropy-chat] Goldbach's Conjecture Resolved! A Story
> 
> The singularity happened all right...
...
> The request was simple. Determine whether the particular number
> 
>     428 08791  39342 94963  22581 56917  17541 83578  75376 65202...
> 33150 27368  53375 70181  59431 66642
> 
> is the sum is the sum of two primes, or not!  ...

I have discovered something truly remarkable.  While fooling with Lee's
question of determining if 4.281E4228 (call it Lee's Number) is the sum of
two primes, I have found a way to determine that Lee's Number can be
expressed as the sum of two primes in approximately 4E3170 different ways.

No kidding, for once.

Let P(n) be the number of different ways a positive even integer n can be
expressed as the sum of two primes.  For consistency with previous work, let
us assume 1 is not prime, but twice a prime counts in P.  For instance,
P(14) = 2 because 11+3 and 7+7 would count but not 13+1.  

I have found that P(n) ~ .15*n^0.75

So P(Lee's Number) ~ 4E3170

This approximation is good to within a factor of two 98.7% of the time and
within a factor of 3 for all even integers above 2 up to 32000, which is as
high as I have checked it.

I found something even more cool that that, which I will post tomorrow.

spike