Abundant and Deficient Numbers
But what about those numbers that are not perfect? How can we call them? Well they are either abundant or deficient. If they are abundant, that means that the sum of their factors is more than their product (or the number itself). If they are deficient, their sum is smaller than their product.
So let us see - you have a number such as 6. Factors of 6 are 1, 2, and 3. Add them up
Look at 12:
Look at 8:
These numbers were first named as such by Nicomachus’ Introductio Arithmetica which was written about 100AD.
Nichomachus identified 6, 28, 496, and 8128 as perfect numbers.
This is a copy of Boethius’ work in philosophy which was written after Nicomachu’s Introductio Arithmetica (c.100AD). It is written in Latin, and was published in Italy c.1390.
See a page on perfect numbers here.
St. Augustine of Hippo (354-430) wrote:
The famous Sir Henry Billingsley's translation of Euclid which appears on these pages, uses a term “perfect number”.
click on the image above to see the web page about this book.
It has been reported by H. Eves in his Mathematical Circles Squared (Boston 1972) that René Descartes said that
The same quote however comes in Samuel Jeake’s Arithmetic (1696) "Perfect Numbers are almost as rare as perfect Men”.