Odd perfect numbers

Abstract

It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10300 (see Brent[1]). Moreover it has been proved by Hagis[4]and Chein[2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in priniciple be obtained solely by calculation, in view of the result of Pomerance[6] who showed that if N is an odd perfect number with at most k prime factors, thenPomerance's work was preceded by a theorem of Dickson[3]showing that there can be only a finite number of such N. Clearly however the above bound is vastly too large to be of any practical use. The principal object of the present paper is to sharpen the estimate (1). Indeed we shall handle odd ‘multiply perfect’ numbers in general, as did Kanold[5], who extended Dickson's work, and Pomerance. Our result is the following.

Maple explorations, perfect numbers, and Mersenne primes

“…Moreover, Hagis [5] has shown (his announcement of this result [6] came a few years earlier in 1975) that an odd perfect number must have at least 8 prime factors. Furthermore, Heath-Brown [7] has shown that if n is an odd perfect number with at most k prime factors, then n < 4'\…”

“…> prevprime(\OO); 97 We can let Maple list for us the first 50 prime numbers: > s:=NULL: for i to 50 do s:=s,ithprime(i)od:s; 2, 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229 Generating Bernoulli numbers:…”

Sieve methods for odd perfect numbers

“…Pomerance [26] gave an effective bound in terms of k. This was improved in succession by Heath-Brown [13], Cook [5], and finally by the second author to N < 2…”

An Upper Bound for the Number of Odd Multiperfect Numbers

“…In 1977, Pomerance [8] gave an explicit upper bound in terms of k. Heath-Brown [5] improved the bound to N < 4 4 k , and Cook [2] reduced this bound to N < D 4 k with D = (195) 1/7 . Nielsen [6] slightly improved and generalised Cook's method; he proved that if N is an odd multiperfect number with k distinct prime factors, then N < 2 4 k .…”