Divisibility, the Fundamental Theorem of Number Theory

Abstract

Counting and the numbers that thus came forth are among the earliest achievements of mankind’s awakening intellect. As numbers came to being, their intriguing properties were revealed, and symbolic meanings were assigned to them. Besides portending fortune or doom, numbers afforded a mathematical expression to many other aspects of existence. For instance, the ancient Greeks considered the divisors of a number that are less than the number itself to be its parts; indeed, they so named them. And those numbers that rise up from their parts, like Phoenix, the bird that according to legend rises up from its own ashes, were viewed as the embodiment of perfection. Six is such a perfect number, since it is the sum of its parts 1, 2, and 3; 28 and 496 are also perfect. Euclid (third century b.c.) already knew that a number of the form 2ⁿ⁻¹(2ⁿ − 1) is perfect if the second factor is prime.1 It was Leonhard Euler (1707–1783), more than two thousand years later, who first showed that any even perfect number must be of this form. To this day it is still unknown whether or not there exist odd perfect numbers.