Ken Ono, left, and Zach Kent.(Photo courtesy of eScienceCommons)
Ken Ono of Emory University and his collaborators have accomplished a breakthrough for so-called partition numbers, a subject of intense mathematical scrutiny for a long time. Initially a very mysterious object, partition numbers are now completely understood in terms of a finite formula and a fractal pattern. These discoveries were the subject of a recent three-day symposium held at Emory.
For every natural number n, we define its partition number, p(n), as the number of ways in which we can break down n as a sum of positive integers. For example,
4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1
Since we can write the number 4 as a sum in five different ways (including the number itself), p(4) = 5.
If we start computing p(n) for larger values of n, we find that these numbers grow very quickly. For instance, p(128) runs to ten digits. Many number-theoretic questions arise naturally from this simple idea of partition number: Can we compute p(n) for any n? How quickly does this sequence grow? Do these numbers follow some sort of divisibility pattern?
Although partition numbers involve nothing more than the simple ideas of adding and counting—mathematicians have had great difficulty in trying to compute and understand these mysterious numbers. As is common in number theory, Ono says, “sometimes the problems which are simplest to the state are the hardest to solve”.
(l-adic) space oddity
Ramanujan also studied the divisibility properties of partition numbers, and he found some interesting congruences. In a 1919 paper, he states:
“I have proved . . . that
There appear to be corresponding properties in which the moduli are powers of 5, 7, or 11 . . . , and no simple properties for any moduli involving primes other than these three.”
In other words, the first congruence says that starting with 4, every fifth partition number is a multiple of five. The curious thing, Ono points out, is that there are no simple properties like the ones mentioned above for any other primes.
But what about the simplest primes: 2 and 3? Silviu Radu recently showed that it’s impossible to find arithmetic progressions of the form An + B for which the partition numbers are always even or always a multiple of three.
In 2000, Ono proved that for all primes Q greater than or equal to 5, there are infinitely many arithmetic progressions of the form An + B for which the partition numbers are always a multiple of Q. However, Ono notes, the congruences that he found are “monstrosities,” like
Most partition numbers don’t fall into Ramanujan’s congruences or Ono’s monster congruences. So, even after Ono’s work, much remained mysterious.
In 1967, A. O. L. Atkin had proved some of Ramanujan’s conjectured “corresponding properties” for the prime 11, by using a clever alternating sequence of operators. Ono started playing with this approach at the American Institute of Mathematics, looking at certain sequences of partition numbers. He began with the sequence
and he noticed that the partition numbers are all multiples of 5. Indeed, ignoring the first term, the rest are all multiples of 52. Eventually, for any m, all the terms in the sequence are divisible by 5m.
Two numbers are said to be 5-adically close if their difference is divisible by a large power of 5. In terms of the number-theoretic picture for 5-adic distances for partition numbers, we’re zooming in closer and closer. So we are looking at a sequence that exhibits the same type of pattern as we zoom to higher resolutions.
This discovery led Ono to think about fractals, which are more often associated with geometric structures than with number-theoretic patterns. Ono wondered if other primes exhibited such a self-similar pattern. Using results by Atkin, he soon found it for the prime 13. In this case, self-similarity means that terms in the sequences mod some power of 13 can be found from the previous term in the sequence. In other words, the sequences are periodic with period 1.
The theorem proved by Folsom, Kent, and Ono simply states that for any prime number, the partition numbers are l-adically fractal (these particular sequences exhibit a periodicity that doesn’t change when we zoom in to the l-adic picture), and gives an upper bound for their Hausdorff dimension.
Magically, Ramanujan’s congruences for 5, 7, and 11 fall right out of this new theorem. The reason these are the only primes with “simple congruences” is that they are the only primes that give a 0-dimensional fractal structure to the partition numbers.
For better understanding –
Video Courtesy- “Numberphile“