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.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.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.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 thesis.degree.grantor University of Waikato thesis.degree.level Doctoral thesis.degree.name Doctor of Philosophy (PhD) en_NZ dc.date.updated 2010-07-01T22:51:52Z pubs.place-of-publication Hamilton, New Zealand en_NZ
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