p(n) is a fractal when n is prime

This Friday night, according to a publicity release by Emory University, math professor Ken Ono will show that partition numbers of primes are fractals.

A partition number is the number of (different) sums of positive integers that produce a specific natural number. Each of the sums is called a partition

For example, the partitions of the number 8 are:

8
7 + 1
6 + 2
6 + 1 + 1
5 + 3
5 + 2 + 1
5 + 1 + 1 + 1
4 + 4
4 + 3 + 1
4 + 2 + 2
4 + 2 + 1 + 1
4 + 1 + 1 + 1 + 1
3 + 3 + 2
3 + 3 + 1 + 1
3 + 2 + 2 + 1
3 + 2 + 1 + 1 + 1
3 + 1 + 1 + 1 + 1 + 1
2 + 2 + 2 + 2
2 + 2 + 2 + 1 + 1
2 + 2 + 1 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Counting up these different partitions yields 22. (A partition is not different when the addends are differently arranged. 1 + 3 is the same partition as 3 + 1. Remember the associative property?) Thus, the natural number 8 thus has a partition number of 22. Or as expressed in number theory p(8) = 22.

Partitions can be illustrated as Young diagrams. The illustration on the left shows Young tableaux of partition numbers making up each of the integers from 1 to 8. The diagrams are arranged in a sort of Russian doll showing the “conjugate partitions.” This diagram is a Ferrers diagram named for Norman Macleod Ferrers who discovered the conjugacy of partitions. I will leave you to Wikipedia’s explanation of the Ferrers diagram.

Partition numbers rapidly increase. You can see yourself:

p(3) = 3
p(4) = 5
p(5) = 7
p(6) = 11
p(7) = 15

p(10) = 42

p(100) = 190,569,292

p(200) = 3,972,999,029,388

You can generate your own set of partition numbers by an on-line calculator devised by Henry Bottomley.

Evidently to increase interest in the lecture the Emory University press release gives no more details to the discovery than a description of the Eureka moment Ono and a colleague had during a hike in a forest when the visualized the patterns as clumps of trees. The press release describes the discovery  by saying that the researchers: “have unlocked the divisibility properties of partitions, and developed a mathematical theory for ‘seeing’ their infinitely repeating superstructure. And they have devised the first finite formula to calculate the partitions of any number.”

If this work is ever published, I’ll update this post here.

Quick Update: The paper entitled Jan Hendrick Bruinier and Ken Ono, “An Algebraic Formula for the Partition Function” is available in pdf format here. The paper concerning the fractal nature of partition numbers of prime, Amanda Folsom, Zachary A. Kent and Ken Ono, “l-adic Properties of the Partition Function” is available in pdf format here.

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