24 (number) - in Mathematics

In Mathematics

• 24 is the factorial of 4 and a composite number, being the first number of the form, where 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.
• 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
• Subtracting 1 from any of its divisors (except 1 and 2, but including itself) yields a prime number; 24 is the largest number with this property.
• 24 has an aliquot sum of 36 and the aliquot sequence (24, 36, 55, 17, 1, 0).
• The aliquot sum of only one number, 529 = 232, is 24.
• 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.
• 24 is the sum of the prime twins 11 and 13.
• 24 is a Harshad number.
• 24 is a semi-meandric number.
• 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.
• The tesseract has 24 two-dimensional faces (which are all squares).
• 12+22+32+...+242 =702 is a perfect square; 24 is the only integer greater than 1 with this property.
• 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 densest lattice sphere-packing problem 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 12+22+32+...+242 =702.)
• The modular discriminant Δ(τ) is proportional to the 24th power of the Dedekind eta function η(τ): Δ(τ) = (2π)12η(τ)24.
• The Barnes-Wall lattice contains 24 lattices.
• The divisors of 24 — namely, {1, 2, 3, 4, 6, 8, 12, 24} — are exactly those n for which every invertible element x of the commutative ring Z/nZ satisfies x2 = 1. Thus the multiplicative group (Z/24Z)× = {±1, ±5, ±7, ±11} 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 analogue in any other dimension. Its 24 vertices can be expressed as the set {±1, ±i, ±j, ±k, (±1 ± i ± j ± k)/2} of unit quaternions, using all choices of signs. This set forms a group under quaternion multiplication, isomorphic to the binary tetrahedral group. The 24-cell tiles 4-dimensional space.
• 24 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can touch another one without overlapping. (The centers of such 24 spheres form the vertices of a 24-cell.)
• For any prime greater than 3, is divisible by . Hence, any square of a prime > 3 expressed in radix 24 ends with an 1. For example, 22 = 4₂₄ and 32 = 9₂₄, but 52 = 11₂₄, 72 = 21₂₄, 112 = 51₂₄, 132 = 71₂₄, etc.
• 24 is the second Granville number, the previous being 6 and the next being 28. It is the first Granville number that is not also a conventional perfect number.
• 24 is the largest integer that is evenly divisible by all natural numbers no larger than its square root.