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

Anders Sandberg asa at nada.kth.se
Thu Oct 12 21:08:55 UTC 2006

spike wrote:
> Let P(n) be the number of different ways a positive even integer n can be
> expressed as the sum of two primes.
> I have found that P(n) ~ .15*n^0.75

Is that a numerical estimate?

http://homepage.mac.com/billtomlinson/primes.html has a neat graph of
P(n). It seems that numbers divisible by different small primes have their
own "paths", I wonder if they all have 0.75 as exponent?

Apparently P(n) is called the Goldbach partition, and it is sequence
A001031  in the encyclopedia of integer sequences. But according to
P(n) has the form C (n/(log(n)log(n-2))) prod_{p in odd prime factors of
n}((p-1)/(p-2)) - which seems to grow roughly slightly faster than
(assuming there are log(log(n)) prime factors
and their size grows as n

suggests that it is bounded by exp(alpha x^beta) alpha>0, 0<beta<1, which
ought to grow much faster.

In any case the partition function turns out to have some cool fractal

Fun! Thanks, Spike for bringing it up!

Anders Sandberg,
Oxford Uehiro Centre for Practical Ethics
Philosophy Faculty of Oxford University

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