Six is Perfect

Although the number seven might be considered lucky, today’s episode is all about the number six. According to the ancient Christian scholar Augustine, God could have created the world in an instant but chose to do it in a perfect number of six days.  Quran also asserts that Allah created the heavens and the earth, and all that is between them, in six days (7:54). The length of these days is, however, not precisely defined. The word “youm” in the verse is understood within the Qur’an to be a long period of time – an era, or an epoch. Whatever be the significance ascribed to it, the search for perfect numbers was behind some of the greatest discoveries in number theory. Since the dawn of mathematics, people have been fascinated with the patterns that can be found in the number system.

A perfect number is a positive integer that equals the sum of its proper divisors, excluding the number itself. For example, 6 is a perfect number because the proper divisors of 6 are 1, 2 and 3 and 6=1+2+3.  This is also known as its aliquot sum. Other perfect numbers are 28, 496, and 8,128. Sometimes these are called ideal or complete numbers. The discovery of such numbers is lost in prehistory. The early Jewish commentators felt that the perfection of the universe was shown by the moon’s period of 28 days. It is not known when the perfect numbers were first studied which might go back to the earliest times when numbers first aroused curiosity.  It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally, given the way their methods of calculations would work. Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties. Ancient Greeks had also studied these, for mystical reasons. The number 6 stands for completeness and symbolizes beauty and high ideals. Six is also the atomic number for carbon, the element of life. It also symbolises Venus, the goddess of love. The standard flute has six holes and so does the standard guitar, six strings. Insects have six legs and the cells in the beehive honeycomb are six-sided. There are six players on a volleyball team and an ice hockey team. We are thrilled by a sixer in a cricket match. Six is also the highest number on the dice. There is also something called “the sixth sense” which refers to extra sensory perception. It is common to use this phrase when we mean hunch or instinct.

It is somewhat surprising that many elementary questions about perfect numbers are still open. Most notably, it is not known whether there are infinitely many perfect numbers, and it is not known whether there are any odd perfect numbers. All even perfect numbers have a very precise form while as the odd perfect numbers either do not exist, or are rare. According to Sylvester, a prolonged meditation on the subject of existence of odd perfect numbers leads us to be satisfied that it would be little short of a miracle. Currently, the largest known perfect number is an even number with 148,414,561 binary digits.  When written in decimal, every even perfect number ends in the digits 6 or 28. The iterative digital sum of any even perfect number is 1 with the exception of 6. The only square-free perfect number is 6.

The Arab mathematicians were also fascinated by perfect numbers. Ibn al-Haytham in his unpublished work, Treatise on analysis and synthesis, showed that perfect numbers satisfying certain conditions had to be of the form 2^{k-1}(2^{k} – 1) where 2^{k} – 1 is prime.  Similarly, Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined, that the numbers of the form 2^{n}p, where p is prime, can be perfect. Yet another Arab mathematician to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239). Other distinguished mathematicians who made a significant contribution to the study of perfect numbers were; Edouard Lucas, Pierre de Fermat  and Marin Mersenne of France,  Leonhard Euler of Sweden, Euclid of Alexandria, J. J. Sylvester of England,  Frank Nelson Cole of America, Pietro Cataldi of Italy, Nicomachus of Gerasa and may be many more.

Remarkably, human existence also manifests itself in six demarcated levels. These six intervals can be briefly put as: The 1st is the pre-existence of the human souls who were destined to be born in this world. The 2nd is this world itself where we live and wherefrom the death may lift us some day. The 3rd is the world of the lesser resurrection or grave. The 4th is the world of the greater resurrection or doomsday. The 5th is the world of heaven and hell, in which the acts and their consequences return to their owners and only God’s mercy can provide some relief. The 6th world is a place elevated above paradise where there shall be an intimate proximity to God granted to those with special insight in reward to performing the most cherished duties towards their Lord.