The Mathematical Mystery of Odd Perfect Numbers: An Exploration of the Odd Perfect Number Problem, Multiply Perfect Numbers, and Related Topics in Number Theory
Item
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Title
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The Mathematical Mystery of Odd Perfect Numbers: An Exploration of the Odd Perfect Number Problem, Multiply Perfect Numbers, and Related Topics in Number Theory
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Creator
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Jasmine Hiebert
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Date of Award
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6/15/2025
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Honors Program Director
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Gavin Keulks
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Faculty Advisor
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Cheryl Beaver
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Abstract
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A perfect number is a number that is equal to the sum of its proper divisors.
For example, 6 is a perfect number because its proper divisors are 1, 2, and
3 and 6 = 1 + 2 + 3. A non-example of a perfect number is 8, whose proper
factors sum to 1 + 2 + 4 = 7 , 8. The next smallest perfect number after 6 is
28 = 1+2+4+7+14 and after 28 we have 496. There are 52 known perfect
numbers. The largest one was discovered in October 2024 and has more than
82 million digits ([10]).
All 52 known perfect numbers are even. Given this observation, it is natural
to ask, do odd perfect numbers exist? Indeed, many people have asked this
question before. Number theorists have been wondering whether or not odd
perfect numbers exist for more than 2,000 years and yet we remain unable to
rigorously answer the problem. Our understanding of odd perfect numbers
provides a stark contrast to how well-studied and formulated even perfect
numbers are. Every few years increased computational power partnered with
our understanding of even perfects allows us to find a new even perfect number.
The more we search for odd perfect numbers, the more elusive they
seem.
We will be talking about the odd perfect number problem and a smorgasbord
of related problems, such as multiply perfect numbers, super perfect
numbers, and primitive abundant numbers in our literature review. Before
we do so, we will formally define all the terms and functions we need in the
Theoretical Frameworks section, which can be used as a reference for these
terms as we proceed through the rest of the paper. Our discussion will culminate
in my original work with non-divisors of odd perfect numbers and
determining the possible forms of multiply perfect numbers.
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Type
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Text
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Department
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Honors
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Language
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eng
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Rights
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Western Oregon University Library has determined, as of 6/29/2025, this item is in copyright, which is held by the author. Users may use the item in accordance with copyright limitations and exceptions, including fair use. For other uses, please ask permission from the author.
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https://rightsstatements.org/page/InC/1.0/?language=en
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Identifier
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honorstheses_091725a
