An Estimation of the Proportion of Abundant Numbers
Document Type
Oral Presentation
Campus where you would like to present
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