#### Title

An Estimation of the Proportion of Abundant Numbers

#### Document Type

Oral Presentation

#### Location

SURC 202

#### Start Date

16-5-2013

#### End Date

16-5-2013

#### Abstract

For 2,500 years mathematicians have studied the “Sum of Divisors” function σ(n). To give an example, σ(12) = 28 because the divisors of 12 are 1, 2, 3, 4, 6 and 12, and the sum of the divisors of 12 is given by 1+2+3+4+6+12 = 28. Now depending on the value of σ(n) – n we classify these numbers as either deficient, perfect or abundant. Perfect numbers are numbers for which σ(n) – n = n, deficient numbers are numbers for which σ(n) – n < n and abundant numbers are numbers for which σ(n) – n > n. The question I am asking is: what is the proportion of abundant numbers? The best current bounds for this proportion are .2476171 and .2476475. I have written code in Java and used it to generate data to try to improve this estimation using statistics. This presentation will cover what an abundant number is, how I generated my data, how I used statistics to try to close the gap for the estimation of the proportion of abundant numbers, and what the result were.

#### Recommended Citation

Pidde, Melissa, "An Estimation of the Proportion of Abundant Numbers " (2013). *Symposium Of University Research and Creative Expression (SOURCE)*. 87.

https://digitalcommons.cwu.edu/source/2013/oralpresentations/87

#### Additional Mentoring Department

Mathematics

An Estimation of the Proportion of Abundant Numbers

SURC 202

For 2,500 years mathematicians have studied the “Sum of Divisors” function σ(n). To give an example, σ(12) = 28 because the divisors of 12 are 1, 2, 3, 4, 6 and 12, and the sum of the divisors of 12 is given by 1+2+3+4+6+12 = 28. Now depending on the value of σ(n) – n we classify these numbers as either deficient, perfect or abundant. Perfect numbers are numbers for which σ(n) – n = n, deficient numbers are numbers for which σ(n) – n < n and abundant numbers are numbers for which σ(n) – n > n. The question I am asking is: what is the proportion of abundant numbers? The best current bounds for this proportion are .2476171 and .2476475. I have written code in Java and used it to generate data to try to improve this estimation using statistics. This presentation will cover what an abundant number is, how I generated my data, how I used statistics to try to close the gap for the estimation of the proportion of abundant numbers, and what the result were.

## Faculty Mentor(s)

Dominic Klyve