The Interesting Properties of 2

The number 2 has some interesting properties which Number Theorists and other Mathematicians might find interesting or ascetically pleasing.

2 is the only even prime number, and the first of all prime numbers. There is no other prime number which is even, due to the parity set of the numbers which constitute the even numbers. In other words, all even numbers are a multiple of 2, and thus will have 2 as a additional factor.

$E = {2n : n \in \mathbb{Z}}$ (1)

The product 2 and the sum 2 is equal.

$2x2 = 2+2$  (2)

The number 2 is the first even number. The exponent 2 is the most common exponent in mathematics. The Goldbach Conjecture states that any even number greater than 2 is the product of two prime numbers.

A natural number is only the sum of two or more consecutive integers if it is not equal to a power of 2.

Fermat’s Last Theorem states that there is no solutions to (3) when $n > 2$

$x^n+y^n=z^n$ (3)

The number 2 is the first prime and even deficient number. A deficient number is any number where the sum of divisors is less than the double of that number.

$\sigma(n) < 2n$ (4)

For example, the divisors for 2 are 1 and 2, the sum of these divisors is 3, and the 2×2 is equal to 4, therefore 2 is considered to be a deficient number. The deficiency number of 2 is 1.

The deficiency number is calculated from the following expression:

$2n - \sigma(n)$ (5)

Descartes proposed that in all simple polyhedra, the sum of the number of vertices and the number of faces  is greater than the number of edges by 2.

All convex polyhedra have an Euler Characteristic (6) of 2.

$\chi=V-E+F$  (6)

V – number of vertices, E – number of edges and F – number of faces.

The number 2 is primorial, and is the second primorial number.

A primorial number is defined mathematically as the following:

$p_{n}\#=\prod_{k=1}^{n}p_{k}$ (7)

where $p_{n}$ is the n-th prime number, and thus the n-th primorial is therefore the product of the first n primes.

The number is also it’s own factorial:

$2! = 2$ (8)

2 is a Motzkin number and a Motzkin Prime. A Motzkin Prime is simply a Motzkin number which is additionally prime.

A Motzkin number is defined as the number of ways to draw non intersecting chords on a circle between n points.

The number 2 is a Meandric number, Open Meandric number and Semi-Meandric number.

A Meandric number is defined as a number where a meander (closed non-self-intersecting curve in two dimensions) intersects a line at 2n points.

References:

Meander (mathematics) – Wikipedia

2 (number) – Wikipedia

The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition) – David Wells page(s) 24