St. Augustine was born in 354 AD, in Thagaste (now Souk Ahras), in Numidia; and he died Bishop of Hippo (in Numidia) in 430. As the most important of the church fathers, he built the theoretical foundation for the Roman Church. His most important works are "De civitate Dei" (412-426), and "Confessiones" (397-400) - in English known as "City of God", and "Confessions". He claimed that God created the world in 6 days, because 6 is a perfect number, and it would have been perfect even without the creation.

Jewish scholars saw the moon's period of 28 days as a reflection of the perfection of the universe. The number 28 is also perfect.

The concept of perfect numbers has been known for millennia, and the first formula for finding them is attributed to Euclid (about 325-265 BC). According to this all even perfect numbers can be calculated, although that was not proved until the 18th century, by Leonhard Euler (1707-1783). During the time in between them, several mathematicians took interest in this subject, most notably: Nicomachus of Gerasa (about 60-120 AD), Thabit ibn Qurrah (836-901), Ibn al-Haytham (965-1039), Ismail ibn Ibrahim ibn Fallus (1194-1239), Regiomontanus (1436-1476), Pietro Antonio Cataldi (1548-1626), René Descartes (1596-1650), Marin Mersenne (1588-1648), and Pierre de Fermat (1601-1665).

Odd ones are still unknown, and it is not known if odd perfect numbers can exist.

A perfect number is an integer that can be expressed as the sum of its own (aliquot) divisors (not including the number itself). The divisors of 6 are 1, 2, and 3; the sum of these is 6, the number itself. Thus 6 is a perfect number.

The next one is 28; which has the divisors: 1, 2, 4, 7, and 14. 1+2+4+7+14=28. So 28 is a perfect number.

The third one is 496, the forth is 8128, the fifth is 33550336, and the sixth is 8589869056.

The size of the numbers increases rapidly. The 24th one consists of 12003 digits!

Euclid wrote in "The Elements":

"If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect."

1+2+4=7 (2 = 2 times 1, 4 = 2 times 2, that is what Euclid means by "double proportion". 7 is a prime. The sum is 7, the last is 4, 7 times 4 equals 28, which is the second perfect even number. Interested readers can build further on this.

The same method was differently expressed by Nicomachus:

"There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totalling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity."

There is good reason to believe that none of these gentlemen invented the method, which probably was a heritage from one of the older high cultures.

Multiperfect Numbers

An integer is called multiperfect if the sum of its divisors, including the integer itself, is a multiple of the original integer.

The first multiperfect number is 120. Its aliquot divisors are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, and 60. The sum of these is 240. Then we should add the original integer, 120, and we get 360. Since that is three times 120, 120 is a triperfect number.

In a similar manner, one can find numbers that are quadriperfect, quinqueperfect, sexiperfect, septiperfect, and octiperfect.

Perfect numbers are really biperfect, but that term is rarely used, since one normally includes in the definition of a perfect number that its divisors should not include the number itself - while in multiperfect numbers one does include it.

(This material has previously been published in TMA/Meriondho Leo and in my e-book “Numericon”.)

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