Perfection

Pythagoras is one of the most interesting and puzzling men in history. He founded a religion ,of which the main tenets were transmigration of the souls and the sinfulness of eating beans. But the unregenerate hankered after beans, and sooner or later rebelled.

- Bertrand Russell, A History of Western Philosophy

Six is a number perfect in itself, and not because God created all things in six days; rather the universe is true; God created all things in six days because this number is perfect. And it would remain perfect even if the work of the six days did not exist.

- St. Augustine

Just as the beautiful and the excellent are rare and easily counted, but the ugly and the bad are prolific, so also abundant and deficient numbers are found to be very many and in disorder, their discovery being unsystematic. But the perfect are both easily counted and drawn up in a fitting order.

- Nichomachus, A.D. 100

वैसे भी परफेक्शन को सुधारा नही जा सकता है|

- आमिर खान ने कहा 'दिल चाहता है' इस फ़िल्ममें

'Perfection' is what people like me crave for, within myself and also things around. Last night a meditation(caused due to it's study) provoked me to think about the perfect example for 'Perfection' in terms of Mathematics. A quick search once again yielded a garden that is overwhelming. My own study provides multitude of inputs about the illustration of 'Perfection' in Pure Mathematics, well known to mathematicians as 'Perfect Numbers', a class of numbers.

Perfect Numbers by definition are the sum of their divisors.

The best and the first easiest example is the number 6.
6 is a very easy example of the definition.
Divisors of 6 are 1, 2, 3
Sum of Divisors - 1 + 2 + 3 = 6

Next perfect number is 28.
Divisors are 1,2,4,7,14.
Sum - 1 + 2 + 4 + 7 + 14 = 28

Guess who discovered them? Our very own Pythagoras.
Quiet naturally the numbers which are not perfect are called Imperfect Numbers.
Pythagoras considered perfect numbers as divine and imperfect numbers as evil.

Another class of numbers are Abundant(where the original number is less than the sum of its factors. Eg - 12(1,2,34,6 are its factors and 1+2+3+4+6=16))

and

Deficient(where the original number is greater than the sum of its factors. Eg - 8(1,2,4 are its factors and 1+2+4=7)).

Most numbers are either Abundant or Deficient. Perfection is rare.

Another class are Amicable Numbers!

Two number are said to be Amicable or friendly if the sum of the divisors of the first number is equal to the second number, & vice versa. An example is of 220 & 284.

220 - factors(1,2,4,5,10,11,20,44,55,110); their sum = 284

284 - factors(1,2,4,71,142); their sum = 220

Further details are in the below references which also shows many other Amicable Numbers. Pythagoras considered 220 & 284 are a divine and are a perfect marriage couple.

Possibly my readers can now understand 'Numerology' which just means the logic behind Numbers how it got its importance into pseudo science such as Astrology, just like Astronomy got into. Pythagoreans created Numerology who thought numbers are divine. Beyond that there is no other logic.

Continuing with the discovery of class of numbers are Sociable Numbers.

Social Numbers are sets of number where the sum of divisors of each number is the next number of a chain. Here is a chain:

12, 496 --> 14, 288 --> 15, 472 --> 14, 264 --> 12, 496

The chain always return to the staring number.

Coming back to Perfect Numbers here is the list of first 10 perfect numbers:

1. 6
2. 28
3. 496
4. 8128
5. 33550336
6. 8589869058 (Discovered in 1588 by Cataldi)
7. 137438691328 (Discovered in 1588 by Cataldi)
8. 2305843008139952128 (Discovered in 1772 by Euler)
9. The one discovered by Russian Mathematician Pervushin in 1883
   
10. The one discovered by Powers in 1911. Sorry folks, citation required

The 30th one was computed by the Cray Super Computer in 1985.

Euclid proved a rule to express Perfect Numbers -

2^X(-1+2^(X+1)) for special values of X.

Odd Perfect Numbers
Odd perfect numbers are even more fascinating than even ones for the sole reason that no-one knows if odd perfect numbers exist. They may remain forever shrouded in mystery. On the other hand, mathematicians have cataloged a long list of what we could be known about odd perfect numbers; for example, it is believed that none will be discovered with values less than 10^(200). Mathematician Albert Beiler says, "If an odd perfect number is ever found, it will have to have met more stringent qualifications than exist in a legal contract, and some almost as confusing.

An odd perfect number

1. must leave a remainder of 1 when divided by 12 or a remainder of 9 when divided by 36.
2. must have at least six different prime divisors.
3. if not divisible by 3, must have at least 9 different prime divisors.
4. if less that 10^(9118), is divisible by the 6th power of some prime.

Author and mathematician David Wells comments, "Researchers, without having produced any odd perfects, have discovered a great deal about them, if it makes sense to say that you know a great deal about something that may not exist."

What Judaism, Christianity and Islam think about Perfect Numbers...

• Ancient Jews tried to use numbers to prove to atheists that the Old Testament was part of God's revelation. Philo justified the story of Genesis by first validating the assertion that God created the world in six days. He claimed that the 6-day creation must be correct because 6 led to it becoming the symbol of creation and reinforcing the notion of God's existence. Despite the fact that the Jews scoffed at the Greek legends and numerology, they began to use a Pythagorean concept to reinforce the notion that Jahweh was the one true God.
• Christianity got interested in 12th Century especially in the second perfect number - 28. For example, since the lunar cycle is 28 days, and because 28 is perfect, philosopher and theologian Albert Magnus (AD 1200-1280) expressed the idea that the mystical body of Christ in the Eucharist appears in 28 phases.
• In Islam, 28 letters of the alphabet in which the Koran is written with the 28 "lunar mansions". It also names 28 prophets before Mohammed.

References:

• Amicable Numbers
• Perfect Numbers
• Social Numbers
• Pervushin
• Cray
• My own set of searches in libraries and book shops.
• Inter net's many other resources