Title

An Estimation of the Proportion of Abundant Numbers

Presenter Information

Melissa Pidde

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.

Faculty Mentor(s)

Dominic Klyve

Additional Mentoring Department

Mathematics

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May 16th, 1:30 PM May 16th, 1:50 PM

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.