Perfection - Perfect Numbers

Perfect Numbers

Perfect numbers have been distinguished ever since the ancient Greeks called them "teleioi." There was, however, no consensus among the Greeks as to which numbers were "perfect" or why. A view that was shared by Plato held that 10 was a perfect number. Mathematicians, including the mathematician-philosopher Pythagoreans, proposed as a perfect number, the number 6.

The number 10 was thought perfect because there are 10 fingers to the two hands. The number 6 was believed perfect for being divisible in a special way: a sixth part of that number constitutes unity; a third is two; a half — three; two-thirds (Greek: dimoiron) is four; five-sixths (pentamoiron) is five; six is the perfect whole. The ancients also considered 6 a perfect number because the human foot constituted one-sixth the height of a man, hence the number 6 determined the height of the human body.

Thus both numbers, 6 and 10, were credited with perfection, both on purely mathematical grounds and on grounds of their relevance in nature.

Belief in the "perfection" of certain numbers survived antiquity, but this quality came to be ascribed to other numbers as well. The perfection of the number 3 actually became proverbial: "omne trinum perfectum" (Latin: all threes are perfect). Another number, 7, found a devotee in the 6th-century Pope Gregory I (Gregory the Great), who favored it on grounds similar to those of the Greek mathematicians who had seen 6 as a perfect number, and in addition for some reason he associated the number 7 with the concept of "eternity."

The Middle Ages, however, championed the perfection of 6: Augustine and Alcuin wrote that God had created the world in 6 days because that was the perfect number.

The Greek mathematicians had regarded as perfect that number which equals the sum of its divisors that are smaller than itself. Such a number is neither 3 nor 7 nor 10, but 6, for 1 + 2 + 3 = 6.

But there are more numbers that show this property, such as 28, which = 1 + 2 + 4 + 7 + 14. It became customary to call such numbers "perfect." Euclid gave a formula for (even) "perfect" numbers:

N_p = 2p−1 (2p − 1)

where p and 2p − 1 are prime numbers.

Euclid had listed the first four perfect numbers: 6; 28; 496; and 8128. A manuscript of 1456 gave the fifth perfect number: 33,550,336. Gradually mathematicians found further perfect numbers (which are very rare). In 1652 the Polish polymath Jan Brożek noted that there was no perfect number between 104 and 107.

Despite over 2,000 years of study, it still is not known whether there exist infinitely many perfect numbers; or whether there are any odd ones.

Today the term "perfect number" is merely historic in nature, used for the sake of tradition. These peculiar numbers had received the name on account of their analogy to the construction of man, who was held to be nature's most perfect creation, and above all on account of their own peculiar regularity. Thus, they had been so named on the same grounds as perfect objects in nature, and perfectly proportioned edifices and statues created by man; the numbers had come to be called "perfect" in order to emphasize their special regularity.

The Greek mathematicians had named these numbers "perfect" in the same sense in which philosophers and artists used the word. Jamblich (In Nicomachi arithmeticam, Leipzig, 1894) states that the Pythagoreans had called the number 6 "marriage," "health," and "beauty," on account of the harmony and accord of that number.

The perfect numbers early on came to be treated as the measure of other numbers: those in which the sum of the divisors is greater than the number itself, as in 12, have — since as early as Theon of Smyrna, ca. 130 A.D. — been called "redundant" (Latin: redundantio), "more than perfect" (plus quam perfecti), or "abundant numbers", and those the sum of whose divisors is smaller, as in 8, have been called "deficient numbers" (deficientes).

Currently 47 perfect numbers have been identified.