Title

A Confidence Interval for the Density of Abundant Numbers

Presenter Information

Melissa Pidde

Document Type

Oral Presentation

Location

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.

Faculty Mentor(s)

Klyve, Dominic

Additional Mentoring Department

Mathematics

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May 15th, 8:30 AM May 15th, 8:50 AM

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.