During the summer of 2001, I spent seven weeks at the Mathematics Research Experience for Undergraduates at Rose-Hulman Institute of Technology. The eight participants were split into two research groups: Partition Theory and Hyperbolic Tiling. Instead of being held to a rigorous schedule we were given intellectual independence. This freedom encouraged creativity and originality in our research. In this way the experience emulated research environments for doing a doctoral thesis. The REU has been a very influential component in my mathematical development. At the completion of the seven weeks I had co-authored a paper entitled Congruencies of Restricted Partition Functions. I will give a brief summary of the concepts involved in this article.

The essence of Number Theory is finding simple relationships that encapsulate complex properties of the integers. Amazingly enough, one of the simplest ideas in mathematics, writing a number as a sum or other numbers, can help illuminate the underlying structure of the integers. This is called partition theory. A partition of a nonnegative integer n is a representation of n as a sum of positive integers, called summands, the order of which are irrelevant. The number of these partitions for an integer n is denoted by the partition function, p(n).

The generating function, P(q), for the partition function is

This general partition function, P(n), has been studied extensively, most notably by Ramanujan who discovered the following congruencies:

In order to discover similar congruences we can consider alterations to the partition functin. Namely we can define a restricted partition function as p_k(n), which is the partition of an integer into summands none of which are divisible by k.

The generating function of p_k(n) is:

To see how p(n) compares to p_k(n) let’s consider an example where k=2. We want to find p(4) and p₂(4). To find p(4) we simply find the ways of writing 4 as a sum of other integers. We can compare this with p₂(4) by eliminating those partitions that contain even summands. This is seen explicitly below, with p(4) as the first column and p₂(4) as the second.

We see that p(4)=5 and p₂(4)=2. This brief introduction to the basic concepts of partition theory gives an impression of the ideas involved. When doing more complicated analysis the generating functions provide the basis for most proofs, but there is not space to go into that here. In our paper we found and proved congruencies for p ₃(n) mod 3, p₅(n) mod 5, and p₇(n) mod 7. Specifically, we focused on congruencies where n=q²m+b, with q being prime. Our analysis consisted in seeing which primes lead to nice congruence properties for each restricted partition function, and then proving these congruencies. This led to the following theorems.

Theorem 1    p₃(q² m+b)=0 mod 3, for all natural numbers m, where q is a prime for which q=5,7,11 mod 12 with b satisying q|24b+2.

Theorem 2    p₅(q² m+b)=0 mod 5, for all natural numbers m, where q is a prime for which q=5 mod 6 with b satisying q|24b+2.

Theorem 3    p₇(q² m+b)=0 mod 7, for all natural numbers m, where q is a prime for which q=3 mod 4 with b satisying q|24b+2.

These theorems provide for an infinite number of modular congruencies for each of the three restricted partition functions. The similarity in form of these congruence classes illuminate a common structure shared by all three functions. This suggests the possibility of further generalization to other partition functions. The full proofs of these theorems can be found in our paper, which can be downloaded from www.angelfire.com/realm/mculek.

Matt Culek

(Editor^'s note: Matt is currently busy applying to graduate school.)

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