## Partitions

### Description

A partition of a positive integer n is a way of writing n as a sum of positive integers. The partition number p(n) counts the number of different ways of doing do. For example p(4)=5. Explicitly,

4=4=3+1=2+2=2+1+1=1+1+1+1.

The partition number has many fascinating properties and has been extensively studied throughout history starting (at least) with Euler. In 1918 MacMahon used a recurrence formula to compute p(200) by hand (the number is huge...),
Ramanujan in 1919 discovered some truly remarkable congruences (such as p(5k+4) is always divisible by 5), but also together with Hardy an asymptotic formula for which they employed/developed a now-famous method in complex analysis known as the Hardy-Littlewood circle method. This was later refined by Rademacher which produced a very nice rapidly converging infinite series expression for p(n).
Since then the interest in partition numbers never ceased and in fact in the last 20 years some spectacular progress has been made, in particular by Bringmann, Bruinier, and Ono and their school(s).

In this project we study the partition number starting with Euler and Ramanujan!

### Resources

There are many sources, but for initial reading the wikipedia entry
• https://en.wikipedia.org/wiki/Partition_(number_theory)
should suffice.

### Prerequisites

• Elementary Number Theory II
• Algebra II
• Complex Analysis II

email: J Funke