Tuesday, December 27, 2005

36

The Wikipedia has an interesting entry on the number 36.

36 is the natural number following 35 and preceding 37. But that, of course, is obvious. Unless you're a mathematician, in which case you need proof.

It is the square of 6 as well as a triangular number. As a square number, it can be arranged in the form of a, well, square. As a triangular number it can be arranged as an equilateral triangle.

Amazingly, it is also a 13-gonal number. That means it can be arranged it can be arranged in a polygon of 13 sides.

It is a highly composite number. It means that it is a positive integer which has more divisors than any positive integer below it.

It is a Harshad number. A Harshad number is an integer that is divisible by the sum of its digits in a given number base. In base 10, 36 is divisible by 9, which is the sum of its digits (3+6). Harshad is Sanskrit for "great joy."

It is the sum of a twin prime (17 + 19), and the sum of the cubes of the first three integers.

It is the smallest number with exactly 9 divisors.

It is the smallest number n with exactly 8 solutions to the equation φ(x) = n. This is Euler's phi function. This is the number of positive integers less or equal than n and coprime to n.

It is the largest numeric base that most computer systems support because it exhausts the numerals, 0 - 9, and the letters, A - Z.

It is the atomic number of krypton.

"36" is also the title of the song by System of a Down. The lyrics aren't particularly any good, but I do like a couple of stanzas:

Television in disgrace
Life you get old, it's the race,
Can you break out,
Can you break out,
Will you live at your own pace?


It is the number of living righteous people without whom, according Jewish tradition, the world would end.

Finally, it is how old I am today.