Partition Congruences by the Localization Method

The central premise of my project is the study of addition on the whole numbers using new techniques in pure mathematics and computer algebra. A partition of a positive integer n is a representation of n as a sum of other positive integers. For example, the number 4 has five partitions: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The total number of partitions of a given integer n is denoted p(n). Thus, p(4)=5. The main problem in additive number theory is the study of partitions: how many partitions a given whole number has, how to compute them, and the various properties of p(n) and related functions. The sequence of p(n) for n=1,2,3,4,..., exhibits an apparent randomness: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 57, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792,... What makes this sequence so remarkable is that it has some very surprising and significant arithmetic properties. For example, if we start with the fourth number in this sequence, 5, and we take every fifth number thereafter, we have the following sequence: 5, 30, 135, 490, 1575, 4565, 12310, 31185, 75175, 173525, 386155, 831820, 1741630,... This means that for every whole number n, p(5n+4) must be divisible by 5. This result was discovered by Ramanujan more than a hundred years ago. Upon examining additional data, Ramanujan was able to guess some extremely deep and intricate divisibility properties of p(n). His conjectures were gradually proved, with some small modifications, over the next fifty years. Given how random the sequence for p(n) looks, the structure of its arithmetic properties is incredible. Moreover, these divisibility properties are connected to very different problems in mathematics, including the shapes of certain underlying geometrical objects called Riemann surfaces, and certain symmetric functions called modular forms. Studying these other problems requires an understanding of algebra, analysis, geometry, and topology. This is what makes the study of partitions so important: a deeper understanding of p(n) can potentially give us a better understanding of a very wide range of ideas all across the rest of mathematics. I study p(n), together with a large number of different functions that are very closely related to p(n). In particular, I build computational tools that help us look for divisibility properties in these functions, and I develop new techniques to help us prove these properties. I am working closely with Prof. Peter Paule at the Research Institute for Symbolic Computation of Johannes Kepler University Linz, and with Dr. Michael Schlosser at the University of Vienna. 1

The focus of this project is how whole numbers add together. I showed how to extend and improve how we study divisibility properties for arithmetic functions associated with modular forms. To understand what this means, imagine how many ways that a given whole number can be expressed as a sum of other whole numbers. For example, we can write 4, 3+1, 2+2, 2+1+1, 1+1+1+1. So there are five different ways of writing 4 as a sum of other whole numbers. We give this the shorthand notation p(4)=5. If we look at the sequence p(n) for n = 1, 2, 3, 4, 5, 6, 7, 8,, it starts out in the following way: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575,... These numbers look sort of random. But in 1918 Ramanujan discovered that if you start at p(4)=5, and count every fifth number in this sequence above, we have 5, 30, 135, 490, 1575,... These numbers are all divisible by 5. Moreover, whenever 24n-1 is divisible by 5^k, for any integer k, then p(n) must also be divisible by 5^k. This is an incredible result, and it is not random at all! The reason for these patterns is that p(n) is related to a strange object called a modular form. These are really abstract and infinite objects which have connections with the theory of the prime numbers, as well as to important algebraic objects called elliptic curves. Basically, understanding how these strange divisibility properties appear with functions like p(n) helps us understand modular forms, and how they connect to things like elliptic curves, and prime numbers. This in turn has a huge impact on computational mathematics, cryptography (how we hide passwords to protect our information on the internet), physics (how we can properly compute how atomic particles behave in quantum theory and string theory), and other areas of mathematics, like the Goldbach problem and Fermat's Last Theorem. There are other modular forms that relate to other kinds of functions like p(n), which look random, but which contain really important patterns. Some of these sorts of patterns look just like the divisibility properties for p(n) by powers of 5 that I showed you. However, some of them are much harder to understand! I invented new techniques to prove these hard conjectures. These techniques surprised even a few leading experts in number theory, and I was able to prove things that were believed to be really difficult. I also classified the different divisibility problems in this subject based on how difficult they are to prove. This will help future researchers on solving other really hard problems.