Multiplicities Between Modular Congruences

The field of mathematics that I work in is called number theory. The motivating problems in this subject are usually very easy to understand. However, the solutions can be very difficult. Also, the ideas look very unimportant for everyday life. However, this kind of mathematics underlies much of encryption and cryptography. For example, this field of math makes it much safer for you to access your bank account and emails on the internet. One important motivating problem of number theory lies in what are called partitions. A partition is just an expression of a whole number as a sum of other whole numbers. For example, the number 4 has 5 partitions: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Notice that we count 4 by itself, and that we do not care about the ordering of the terms (3+1 is the same as 1+3). This is a very simple idea. All that you need to know about is whole numbers (1, 2, 3, 4, 5,) and addition. Suppose we list out the number of partitions for each whole number. For the small numbers this is easy. After all, 1 has only 1 partition (1), and 2 has only 2 partitions (2, 1+1). If we go a little further, we find that the numbers in the list grow very fast: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, Notice that there is no obvious pattern in these numbers. It is not clear when they are even or odd, or primes, or squares. However, lets look at the sequence of numbers whose last digit is a 4 or a 9 (so 4, 9, 14, 19, 24, 29). If we just look at the number of partitions for these numbers, we have the sequence: 5, 30, 135, 490, Each number is divisible by 5. Lets call this property a congruence. In fact, the partition numbers have similar congruences for 5, 25, 125, and indeed for all powers of 5. They also exhibit similar congruences for powers of 7 and 11. It turns out that partition numbers are not the only numbers that have congruences like this. There are many, many other interesting number sequences which behave in the following way: they look random, but upon closer inspection we find these highly ordered congruences. Moreover, sometimes these properties are very easy to study, and sometimes they are extremely difficult. There are still some today that we have guessed but not yet proved. We do not fully understand why these congruences appear, or why sometimes they are so difficult to study. We do know that they are connected to some of the most difficult problems in number theory. We have recently learned that these congruences can sometimes jump from one sequence of numbers to another. What this means is that a sequence of numbers can have a congruence just because another quite unrelated sequence of numbers has a similar congruence. This tends to happen only with the very difficult congruences. Understanding why this happens can help us understand and predict these congruences, and it might have major implications for the entirety of number theory. The goal of my project is to study this strange new behavior.