# Partitions Connected with the Number 8

## Location

Room 2903

## Session Format

Paper Presentation

## Research Area Topic:

Natural & Physical Sciences - Mathematics

## Abstract

A partition is a way that a number can be written as a sum of other numbers. For example, the number 4 has five different partitions: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. A major achievement in number theory was the development of a formula, due to Hardy and Ramanujan, which can efficiently count up the total number of possible partitions of any whole number. Remarkably, this formula involves pi, infinite series, hyperbolic trigonometric functions, and other astonishing complications for so simple an idea. A natural question to ask is whether similar formulas can be developed to count partitions in which the given summands have certain restrictions. We have derived one such formula, which counts partitions in which the even parts have the form 8m+4, and the odd parts have the form 8m+a or 8m-a, where we fix a to be either 1 or 3. For example, for a=1, the number 4 has only two permissible partitions: 4 or 1+1+1+1. Our formula is every bit as intricate and bizarre as the result of Hardy and Ramanujan, and it is exceptionally precise when tested. This talk will be relaxed and accessible to a large audience. We will produce a variety of examples, show (but not derive) our formula, and demonstrate its reliability when subjected to numerical tests. We will also outline the methods used to derive our formula, and the importance those methods have to the rest of mathematics.

## Presentation Type and Release Option

Presentation (Open Access)

## Start Date

4-16-2016 4:00 PM

## End Date

4-16-2016 5:00 PM

## Recommended Citation

Smoot, Nicolas Allen, "Partitions Connected with the Number 8" (2016). *GS4 Georgia Southern Student Scholars Symposium*. 113.

https://digitalcommons.georgiasouthern.edu/research_symposium/2016/2016/113

Partitions Connected with the Number 8

Room 2903

A partition is a way that a number can be written as a sum of other numbers. For example, the number 4 has five different partitions: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. A major achievement in number theory was the development of a formula, due to Hardy and Ramanujan, which can efficiently count up the total number of possible partitions of any whole number. Remarkably, this formula involves pi, infinite series, hyperbolic trigonometric functions, and other astonishing complications for so simple an idea. A natural question to ask is whether similar formulas can be developed to count partitions in which the given summands have certain restrictions. We have derived one such formula, which counts partitions in which the even parts have the form 8m+4, and the odd parts have the form 8m+a or 8m-a, where we fix a to be either 1 or 3. For example, for a=1, the number 4 has only two permissible partitions: 4 or 1+1+1+1. Our formula is every bit as intricate and bizarre as the result of Hardy and Ramanujan, and it is exceptionally precise when tested. This talk will be relaxed and accessible to a large audience. We will produce a variety of examples, show (but not derive) our formula, and demonstrate its reliability when subjected to numerical tests. We will also outline the methods used to derive our formula, and the importance those methods have to the rest of mathematics.