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The scale of perfection

Published online by Cambridge University Press:  14 July 2016

David G. Kendall
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Abstract

A survey is given of the attitude to and use of the concepts of deficient, perfect, and abundant number from the time of Nicomachus of Gerasa to that of David Anakht and Alcuin of York. Alcuin's ‘Letter to Dafnin', the source of an anecdote frequently mentioned in mathematical texts, is included as an appendix.

The statistics of deficient, perfect, and abundant numbers over the range 1–50000 are studied and presented graphically in several novel ways and compared with the work of Davenport and others on the distribution-law for the values of z = σ (n)/(2n).

Some queries are raised concerning observations which ancient writers might have been expected to make; for example, did they notice that about one half of all the even numbers are abundant?

Keywords

ABUNDANT NUMBERSALCUIN OF YORKDAVENPORT DISTRIBUTIONDAVID ANAKHTNOAH'S ARKPERFECT NUMBERSPRIMITIVE ABUNDANT NUMBERS
Type
Part 3 — Mathematics
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 125 - 138
Copyright
Copyright © 1982 Applied Probability Trust 

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References

[1] ALCUIN OF YORK (FLACCUS ALBINUS) (732–804 AD) The most convenient editions of Alcuin's works are: [i] B. Flacci Albini seu Alcuini Opera Omnia (Volumes 100 and 101 of Patrologiae cursus completus ed. Migne, J.-P.) Paris, 1851. (ii) (for the letters) Monumenta Germaniae Historica: Epistolarum IV: Epistolae Karolini Aevi II , ed. Duemmler, E., Berlin, 1895.Google Scholar
[2] Behrend, F. (1932) Über numeri abundantes. Sitzungsber. Preuss. Akad. Wiss. Berlin , 322–328.Google Scholar
[3] Burton, D. M. (1976) Elementary Number Theory. Allyn and Bacon, Boston.Google Scholar
[4] Davenport, H. (1933) Über numeri abundantes. Sitzungsber. Preuss. Akad. Wiss. Berlin , 830–837. (See also The Collected Works of Harold Davenport, Vol. IV. (Academic Press, London, 1977).).Google Scholar
[5] DAVID INVICTUS (C. 480 AD) The Definitions of Philosophy. The russian translation is: David Anakht: Sochineniya , trans. Arevshatyan, S. S., Moscow, 1975. An english version is being prepared by Bridget Kendall in association with R.V. Ambartzumyan, who initiated this project.Google Scholar
[6] Dickson, L. E. (1913) Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Amer. J. Math. 35, 413–422.Google Scholar
[7] Dickson, L. E. (1913) Even abundant numbers. Amer. J. Math. 35, 423–426.Google Scholar
[8] Dickson, L. E. (1913) Theorems and tables on the sum of the divisors of a number. Quart. J. Pure Appl. Math. 44, 264–296. (These three papers by Dickson are to be found in Vols. I and V of The Collected Mathematical Papers of L. E. Dickson (Chelsea, New York, 1975).).Google Scholar
[9] Kendall, Bridget (1980) Problems of translating David the Invincible's Definitions of Philosophy into english. Unpublished paper presented at the David Anakht Jubilee Celebration Meeting, Erevan.Google Scholar
[10] NICOMACHUS OF GERASA (C. 100 AD) The best (english) source for this author is Nicomachus of Gerasa's ‘Introduction to Arithmetic’, trans. D'Ooge, M. L., ed. Robbins, F. E. and Karspinski, L. C., Macmillan, London, 1926.Google Scholar
[11] Salie, H. (1955) Über die Dichte abundanter Zahlen. Math. Nachrichten 14, 39–46.Google Scholar
[12] THEON OF SMYRNA (C. 140 AD) For this writer see Théon de Smyrne: Exposition des connaissances mathématiques utiles pour la lecture de Platon , trans. and ed. Dupuis, J., Paris, 1892.Google Scholar
[13] Wall, C. R., Crews, P. L. and Johnson, D. B. (1972) Density bounds for the sum of divisors function. Math. Comp. 26, 773–777.Google Scholar