Sunday, March 14, 2010


24 (twenty-four) is the natural number following
23 and preceding 25.

24 is the factorial of 4 and a composite number,
being the first number of the form 23q, where q
is an odd prime

It is the smallest number with exactly eight divisors:
1, 2, 3, 4, 6, 8, 12, and 24. It is a highly composite
number, having more divisors than any smaller
number. Adding up all the proper divisors of 24
except 4 and 8 gives 24, hence 24 is a semiperfect

Subtracting one from any of its divisors (except
1 and 2, but including itself) yields a prime number.
24 is the largest number with this property, for to
have this property a number cannot be divisible
by a prime greater than three, nor can it be divisible
by 9 or 16.

24 has an aliquot sum of 36 and the aliquot sequence
(24,36,55,17,1,0). Twenty-four is only the aliquot sum
of one number the square 529.

There are 10 solutions to the equation φ(x) = 24, namely
35, 39, 45, 52, 56, 70, 72, 78, 84 and 90. This is more
than any integer below 24, making 24 a highly totient number.

24 is a nonagonal number. This number is also the sum
of a twin prime (11 + 13). It is a Harshad number and
a semi-meandric number.

Together with the number one, 24 is one of the few
numbers n for which the sum of μ(d)d2 over the
divisors d of n is equal to itself.

The product of any four consecutive numbers
is divisible by 24. This is because, among any
four consecutive numbers, there must be two
even numbers, one of which is a multiple of
four, and there must be a multiple of three.

In 24 dimensions there are 24 even positive definite
unimodular lattices, called the Niemeier lattices.
One of these is the exceptional Leech lattice which
has many surprising properties; due to its existence,
the answers to many problems such as the kissing
number problem and sphere packing are known in
24 dimensions but not in many lower dimensions.
The Leech lattice is closely related to the equally
nice length-24 binary Golay code and the Steiner
system S(5,8,24) and the Mathieu group M24.
One construction of the Leech lattice is possible
because of the remarkable fact that
12+22+32+...+242 =702 is a perfect square;
24 is the only integer greater than 1 with this
property. These properties of 24 are related
to the fact that the number 24 also appears
in several places in the theory of modular
forms; for example, the modular discriminant
is the 24th power of the Dedekind eta function.

The Barnes-Wall lattice contains 24 lattices.

24 is the highest number n with the property
that every element of the group of units
(Z/nZ)* of the commutative ring Z/nZ, apart
from the identity element, has order 2; thus
the multiplicative group
(Z/24Z)* = {1,5,7,11,13,17,19,23} is isomorphic
to the additive group (Z/2Z)3. This fact plays a
role in monstrous moonshine.

The 24-cell, with 24 octahedral cells and 24
vertices, is a self-dual convex regular 4-polytope;
it has no good 3-dimensional analogue.

For any prime p greater than
3, p2 − 1 is divisible by 24 with no remainder.

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