25 March 2005

Srinivasa Ramanujan's mad skilz

In the New Scientist article "Classic maths puzzle cracked at last", they report that a graduate student from the University of Wisconsin has discovered the pattern behind Srinivasa Ramanujan's [Wikipedia] partition function [Wikipedia]. It broke my brain trying to follow the steps of historical discovery within this problem, but it reminded me much of the Fermat story. Some notes to get it straight in my head:

Whole numbers can be represented as partitions consisting of sums. For example, the number 4 has the partitions:

  1. (4)
  2. (3+1)
  3. (2+2)
  4. (2+1+1)
  5. (1+1+1+1)

The number of partitions is given as p(n), so p(4) = 5. Not only that, but any number that ends with a 4 has a number of partitions divisible by 5. I guess that's: p(n) mod 5 = 0, where (n+10) mod 10 = 4. So, for some values there's a relationship between the number of ways you can sum that value and the value itself. That's weird. Butwaitthere'smore. Numbers ending in 9 also have the partition function p(n) mod 5 = 0. Ramanujan grouped numbers with their partition functions and found that, along with groups for 5, groups of partitions appeared for 7 and 11. He called this relationship congruencies.

There's some fancy math-speak to describe this, but I'll just say that: with p(n) mod m = 0, congruences occur where m={5, 7, 11}. You get my point.

In the 1940s, Freeman Dyson [Wikipedia] found a single rule that explains the congruencies for 5 and 7. In the 1980s Ken Ono found that all prime numbers had congruencies--that is, there exists groups of values whose number of partitions is divisible by a prime. Finally, this year, Karl Mahlburg [Wikipedia] (a student of Ono's) discovered the rule covering not only 5 and 7 but all primes and therefore all congruencies. This is a super-rule to the one that Dyson found.

The maths, as usual, is way beyond me, but the organization of it all is striking. (Side-note: Wow, Mahlburg got into Wikipedia based on this recent paper on congruencies.)

[ posted by sstrader on 25 March 2005 at 1:31:47 PM in Science & Technology ]