So here we are thinking about numbers

Happy new year 2009! I am feeling happy we have left 2008 for 8, while being a favorite number in Chinese culture, has never really appealed to me : a fat little man who has no direction except in the face of infinity. 9 however... 3 to the power of 2! and, even better I turned 27 a couple of weeks ago, which of course is 3 to the power of 3, and 3 being my favorite number, well...

Paul Auster personnalises numbers, and puts it beautifully : "... each number has a personnality of its own. A twelve is very different from a thirteen (...) Twelve is upright, conscious, intelligent, whereas thirteen is a loner, a shady character who won't think twice about breaking the law to get what he wants. Eleven is tough, an outdoorsman, who likes tramping through woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what's he told; nine is deep and mystical, a Buddha of contemplation (...)"

Some of this is cultural of course : thirteen is the traitor Judas and ten is the kid at school who always got top marks. But some numbers possess the extraordinary qualities and this is when we leave the biased lands of cultural associations and disappear down the rabbit hole of the study of numbers and their properties, which started with Pythagoras ("numbers are everything") and continues today to fascinate mathematicians and amateurs. Let's go back to twelve, Auster's natural leader. Twelve is a very useful number, as it can be divided by 1, 2, 3, 4 and 6, but that does not make it perfect mathematically. For this, as defined by the Pythagoreans, a number must be equal to the sum of its divisors. Six is the first perfect number as its divisors (1,2,3) add up to 6. Twenty eight is the next as its divisors (1, 2, 4, 7, 14) add up to itself.

Numbers which are equal to a power of two (4, 8, 16, 32, 64 and so on) are never perfect, but the sum of their divisors is always equal to the number minus one. For example : 2 to the power of 2 is 4 but its divisors (1 and 2) add up to 3. 2 to the power of 3 is eight but eight's divisors (1,2 and 4) are equal to 7. 2^4 is 16, whose divisors (1,2,4 and 8) add up to 15. Euclid established a link between perfect numbers and the number 2. It showed that every perfect nmber (6, 28, 496) is the product of a power of 2, multplied by the next power of 2 minus one.

ie 6 = 2^1 x (2^2 -1)
28 = 2^2 x (2^3 - 1)
496 = 2^4 x (2^5 - 1).

A cool derivation on perfect numbers are what Fermat called friendly numbers, of which he discovered 2 pairs: 220 and 284. the sum of 220's divisors add up to 284 and vice versa. In 1636 he discivered the pair 17296 and 18416. Same goes for 1184 and 1210, discovered by a sixteen year old Italian called Paganini, two hundred and fifty years later. Descartes discovered a pair in the 9 million region and Euler compiled a list of sixty two.

And despite 2009 being neither perfect nor friendly, I wish you both and more this year: Happy New Year!