Multiply perfect numbers of low abundancy

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.