Ken Ono cracks partition number mystery

[yuiop]

Hi, has anyone else seen this news item http://www.physorg.com/news/2011-01-math-theories-reveal-nature.html on how to crack partition numbers using fractals?

It came out on Thursday the 20th Jan 2011. They gave a tantalising glimpse and said the full new theory will be revealed on Fri 21 st Jan. It is now Sat 22nd Jan. Has anyone seen the new theory yet? Does anyone understand the significance? Would it be as significant as being able to trivially predict large prime numbers for example?

 

[epsi00]

we are all waiting for the other shoe to drop. My conclusion is that mathematicians need to take more walks in the wilderness to find new ways of tackling old problems. I am convinced that the same method ( fractal ) can be applied to other problems...but it won't be by me.

 

[RobertT]

I wonder what Terence Tao has got to say about this, since he is one of the leading experts on prime numbers.

Surely, I think Tao will be excited with Ono's discovery.

 

[Studiot]

Thank you for the link. It is certainly an active subject.

 

[Raphie]

l-adic properties of the partition function
Amanda Folsom, Zachary A. Kent & Ken Ono
http://www.aimath.org/news/partition/folsom-kent-ono.pdf

 

[Max™]

I was just about to pop in and post that, I wondered what you folks would think of it.

My interest in number theory is fairly recent, so digesting it is beyond me, but it definitely had that "this is important" feel to it.

 

[Raphie]

NEW THEORIES REVEAL THE NATURE OF NUMBERS
Adding & Counting
Ken Ono (with Jan Bruinier, Amanda Folsom & Jack Kent)
A 70 Minute Lecture for a General Audience

 

[chhitiz]

i just read about fractals and they are pretty awesome. i was just wondering- could fractals be used to design complex artificial surfaces that mimic natural ones, like the surface of a coral, for example?

 

[AlephZero]

i just read about fractals and they are pretty awesome. i was just wondering- could fractals be used to design complex artificial surfaces that mimic natural ones, like the surface of a coral, for example?

Yes, that is one of the methods used to generate realistic looking computer graphics for movies and video games.

 

[chhitiz]

if i have a rule, say, z_i+1=az_i²+bz_i+c,
how do i determine what general form of numbers will follow the fractal pattern for this rule.

 

[Raphie]

From Scientific American...

Mathematics' Nearly Century-Old Partitions Enigma Spawns Fractals Solution
Newly discovered counting patterns explain and elaborate cryptic claims made by the self-taught mathematician Srinivasa Ramanujan in 1919
By Davide Castelvecchi | February 8, 2011
http://www.scientificamerican.com/article.cfm?id=mathematics-ramanujan

excerpt # 1
The patterns link certain sequences of p(n) where the n's are separated by powers of any prime number beyond 11. For example, take the next prime up, 13, and the sequence p(6), p(6 + 13), p(6 + 13 + 13) and so on. Ono's formulas link these values with those of p(1,007), p(1,007 + 13^2), p(1007 + 13^2 + 13^2) and so on. The same formulas link the latter sequence with one where the n's come at intervals of 13^3—and so on for larger and larger exponents. (The formulas are slightly more subtle than just saying that the p(n) are multiples of a prime.) Such recurrence is typical of fractal structures such as a Mandelbrot set [see the video above], and is the number theory equivalent of zooming into a fractal, Ono explains.

excerpt # 2
Do Ono et al.'s discoveries have any practical use? Hard to predict, Andrews says. "Often deep understanding of underlying pure mathematics takes awhile to filter into applications." In the past methods developed to understand partitions have later been applied to physics problems such as the theory of the strong nuclear force or the entropy of black holes.

In reference to the above, for anyone interested...

p (6) = 11
p (6 + 13) = 490
p (6 + 13 + 13) = 8349

p (1007) = 31724668493728872881006491578226
p (1007 + 169) = p (1176) = 19314481663345819649385158162679300
p (1007 + 169 + 169) = p (1345) = 5393578994197824268512706677957552625

a(n) = number of partitions of n (the partition numbers)
http://oeis.org/A000041

 

[Goongyae]

>p (6) = 11
>p (6 + 13) = 490
>p (6 + 13 + 13) = 8349

>p (1007) = 31724668493728872881006491578226
>p (1007 + 169) = p (1176) = 19314481663345819649385158162679300
>p (1007 + 169 + 169) = p (1345) = 5393578994197824268512706677957552625

Each p(1007+13^2 x)=p(6+13*77+13^2x) is divisble by p(6+13x). Is that the property referred to?

Note also that
p(6) is divisible by p(1)=1
p(6+13) is divislbe by p(1+13^0) = 2
p(6+13+13) is divisble by p(1+13^0+13^0) = 3
However p(6+13+13+13)=89134 is not divisible by p(1+13^0+13^0+13^0)=5
The pattern does not seem to work backward, i.e. if zeroth powers are used.

 

[Raphie]

String Theory and Partition Numbers
February 12, 2011 by Philip Gibbs

excerpt...
One thing that is not mentioned in all the recent news coverage is the important connection between partition numbers and string theory that is very easy to see even [at] a very basic level. From the theory of musical harmonics you know that a string has vibration modes labelled by integers k, whose frequency is ωk = kalpha for some constant alpha...

More Here...
http://blog.vixra.org/2011/02/12/string-theory-and-partitions-numbers/