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On the largest component of an odd perfect number

Part of: Elementary number theory

Published online by Cambridge University Press:  09 April 2009

Graeme L. Cohen
Graeme L. Cohen
Affiliation:
School of Mathematics Sciences The New South Wales Institute of Technology Broadway, New South Wales 2007, Australia
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Abstract

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It is shown that any odd perfect number has a component greater than 1020.

MSC classification

Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 2 , April 1987 , pp. 280 - 286
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Brauer, A., ‘On the non-existence of odd perfect numbers of form pαq12 q22…qt-12qt4’, Bull. Amer. Math. Soc. 49 (1943), 712–718.CrossRefGoogle Scholar
[2]Cohen, G. L., ‘Appendices for “On the largest component of an odd perfect number”’ (available from the author).Google Scholar
[3]Guy, R. K., Unsolved Problems in Number Theory (Springer-Verlag, New York, 1981).CrossRefGoogle Scholar
[4]Hagis, P. Jr, ‘A lower bound for the set of odd perfect numbers’, Math. Comp. 27 (1973), 951–953.CrossRefGoogle Scholar
[5]Inkeri, K., ‘On the diophantine equation a(xn -1)/(x-1) = ym’, Acta Arith. 21 (1972), 299–311.CrossRefGoogle Scholar
[6]McDaniel, W. L., ‘The non-existence of odd perfect numbers of a certain form’, Arch. Math. 21 (1970), 52–53.CrossRefGoogle Scholar
[7]McDaniel, W. L., ‘On the divisibility of an odd perfect number by the sixth power of a prime’, Math. Comp. 25 (1971), 383–385.CrossRefGoogle Scholar
[8]Muskat, J. B., ‘On divisors of odd perfect numbers’, Math. Comp. 20 (1966), 141–144.CrossRefGoogle Scholar
[9]Nagell, T., Introduction to Number Theory (Wiley, New York, 1951).Google Scholar
[10]Steuerwald, R., ‘Verschärfung einer notwendigen Bedigung für die Existenz einer ungeraden vollkommenen Zahl’, S.-Ber. Math.-Nat. Abt. Bayer. Acad. Wiss. (1937), 68–72.Google Scholar
[11]Tuckerman, B., ‘A search procedure and lower bound for odd perfect numbers’, Math. Comp. 27 (1973), 943–949.CrossRefGoogle Scholar
[12]Tuckerman, B., ‘Odd-perfect-number tree to 1036, to supplement “A search procedure and lower bound for odd perfect numbers”’, (IBM Research Report RC-4695, 1974, copy deposited in UMT file, reviewed Math. Comp. 27 (1973), 1004–1005).Google Scholar