Abstract
The origins of the study of perfect numbers are lost in antiquity, but the concept was clearly recognized well over 2000 years ago and involves such contributors as Euclid, Fermat, Descartes, Mersenne, Legendre, and Euler. The classification of positive integers as perfect, abundant, or deficient is also an ancient idea, one which dates from before A.D. 100. It is the goal of this note to make explicit an idea used by researchers since the seventeenth century (and perhaps before) and reproduce, comment on, and extend their ideas. For the newcomer to these concepts, the basic definitions are as follows: A perfect number is a positive integer which is equal to the sum of its proper divisors (1 + 2 + 3 = 6; 1 + 2 + 4 + 7 + 14 = 28), an abundant number, is one for which the sum of the proper divisors is greater than the number (1 + 2 + 3 + 4 + 6 > 12), and a deficient number has the sum of its proper divisors less than the number (1 + 2 + 5 2N, perfect if a(N) = 2N, and deficient if u(N) < 2N. Certainly the prime 41 is very deficient since the only divisors of 41 are 1 and 41, and 1 + 41 is much less than 2 * 41, whereas 8 is, relatively speaking, just barely deficient since 1 + 2 + 4 + 8 is close to 2 * 8. On the other hand, 360 is very abundant since the sum of its divisors, 1170, is greater than 3 * 360. By such examples we are led to a natural measure of the abundancy or deficiency of numbers.
Published Version
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