Multiply perfect numbers of low abundancy

Zhou, Qizhi

Multiply perfect numbers of low abundancy

dc.contributor.advisor	Broughan, Kevin A.
dc.contributor.advisor	Sorli, Ron
dc.contributor.author	Zhou, Qizhi
dc.date.accessioned	2010-07-13T00:02:10Z
dc.date.available	2010-07-13T00:02:10Z
dc.date.issued	2010
dc.date.updated	2010-07-01T22:51:52Z
dc.description.abstract	The purpose of this thesis is to investigate the properties of multiperfect numbers with low abundancy, and to include the structure, bounds, and density of certain multiperfect numbers. As a significant result of this thesis, an exploration of the structure of an odd 4-perfect number has been made. An extension of Euler’s theorem on the structure of any odd perfect number to odd 2k-perfect numbers has also been obtained. In order to study multiperfect numbers, it is necessary to discuss the factorization of the sum of divisors, in particular for (qe), for prime q. This concept is applied to investigate multiperfect numbers with a so-called flat shape N = 2ap1 · · ·pm. If some prime divisors of N are fixed then there are finitely many flat even 3-perfect numbers. If N is a flat 4-perfect number and the exponent of 2 is not congruent to 1 (mod 12), then the exponent is even. If all odd prime divisors of N are Mersenne primes, where N is even, flat and multiperfect, then N is a perfect number. In more general cases, some necessary conditions for the divisibility by 3 of an even 4-perfect number N = 2ab are obtained, where b is an odd positive integer. Two new ideas, namely flat primes and thin primes, are introduced since these appear often in multiperfect numbers. The relative density of flat primes to all primes is given by 2 times Artin’s constant. An upper bound of the number of thin primes is T(x) less less x log2 x . The sum of the reciprocals of the thin primes is finite.	en_NZ
dc.format.mimetype	application/pdf
dc.identifier.citation	Zhou, Q. (2010). Multiply perfect numbers of low abundancy (Thesis, Doctor of Philosophy (PhD)). University of Waikato, Hamilton, New Zealand. Retrieved from https://hdl.handle.net/10289/4138	en
dc.identifier.uri	https://hdl.handle.net/10289/4138
dc.language.iso	en
dc.publisher	University of Waikato	en_NZ
dc.rights	All items in Research Commons are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.
dc.subject	multiperfect numbers	en_NZ
dc.title	Multiply perfect numbers of low abundancy	en_NZ
dc.type	Thesis	en_NZ
pubs.place-of-publication	Hamilton, New Zealand	en_NZ
thesis.degree.grantor	University of Waikato
thesis.degree.level	Doctoral
thesis.degree.name	Doctor of Philosophy (PhD)	en_NZ