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Odd perfect numbers

Published online by Cambridge University Press:  24 October 2008

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Magdalen College, Oxford
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Extract

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, then

Pomerance'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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 191 - 196
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Brent, R. P.. Improved techniques for odd perfect numbers. Math. Comp. 57 (1991), 857–868.Google Scholar
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