A Confidence Interval for the Density of Abundant Numbers
Document Type
Oral Presentation
Campus where you would like to present
SURC Room 202
Start Date
15-5-2014
End Date
15-5-2014
Keywords
Statistics, Density, Abundant numbers
Abstract
For more than 2,000 years, mathematicians have studied the “sum of divisors” function σ(n). To give an example, since the divisors of 12 are 1, 2, 3, 4, 6 and 12, the sum of the divisors of 12 is σ(12) = 1+2+3+4+6+12 = 28. Depending on the value of σ(n) we can classify an integer n as deficient, perfect or abundant. An integer n is said to be perfect if σ(n) = 2n, deficient if σ(n) < 2n, and abundant if σ(n) > 2n. Notice this makes 12 an abundant number since σ(12) = 28 > 2(12). We notate the proportion of integers which are abundant numbers as dA. The current bounds on the proportion of abundant numbers are .2476171 < dA < .2476475. This presentation will cover how I used statistics to close the gap and give a new estimation of the proportion of abundant numbers.
Recommended Citation
Pidde, Melissa, "A Confidence Interval for the Density of Abundant Numbers" (2014). Symposium Of University Research and Creative Expression (SOURCE). 7.
https://digitalcommons.cwu.edu/source/2014/oralpresentations/7
Additional Mentoring Department
Mathematics
A Confidence Interval for the Density of Abundant Numbers
SURC Room 202
For more than 2,000 years, mathematicians have studied the “sum of divisors” function σ(n). To give an example, since the divisors of 12 are 1, 2, 3, 4, 6 and 12, the sum of the divisors of 12 is σ(12) = 1+2+3+4+6+12 = 28. Depending on the value of σ(n) we can classify an integer n as deficient, perfect or abundant. An integer n is said to be perfect if σ(n) = 2n, deficient if σ(n) < 2n, and abundant if σ(n) > 2n. Notice this makes 12 an abundant number since σ(12) = 28 > 2(12). We notate the proportion of integers which are abundant numbers as dA. The current bounds on the proportion of abundant numbers are .2476171 < dA < .2476475. This presentation will cover how I used statistics to close the gap and give a new estimation of the proportion of abundant numbers.
Faculty Mentor(s)
Klyve, Dominic