[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
http://www.ieeta.pt/~tos/goldbach.html
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
n/(log(n)^2).
(assuming there are log(log(n)) prime factors
http://mathworld.wolfram.com/DistinctPrimeFactors.html
and their size grows as n
http://www.springerlink.com/content/r52233680688x957/#search=%22%22average%20prime%20factor%22%22
)
However,
http://www.primepuzzles.net/puzzles/GoldbachPartitions.pdf#search=%22Goldbach%20Partition%20asymptotics%22
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
properties:
http://arxiv.org/ftp/nlin/papers/0601/0601024.pdf
Fun! Thanks, Spike for bringing it up!
--
Anders Sandberg,
Oxford Uehiro Centre for Practical Ethics
Philosophy Faculty of Oxford University
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