4

Number
$4$ (four) is:


 * The $2$nd square number, and the first square of a prime number:
 * $4 = 2 \times 2 = 2^2 = 1 + 3$


 * The $2$nd tetrahedral number after $1$:
 * $4 = 1 + 3 = \dfrac {2 \left({2 + 1}\right) \left({2 + 2}\right)} 6$


 * The $1$st semiprime:
 * $4 = 2 \times 2$


 * The $2$nd trimorphic number after $1$:
 * $4^3 = 6 \mathbf 4$


 * The $4$th Ulam number after $1, 2, 3$:
 * $4 = 1 + 3$


 * The $3$rd Lucas number after $(2), 1, 3$:
 * $4 = 1 + 3$


 * The $3$rd highly composite number after $1, 2$:
 * $\tau \left({4}\right) = 3$


 * The $4$th highly abundant number after $1, 2, 3$:
 * $\sigma \left({4}\right) = 7$


 * The $3$rd superabundant number after $1, 2$:
 * $\dfrac {\sigma \left({4}\right)} 4 = \dfrac 7 4 = 1 \cdotp 75$


 * The $3$rd almost perfect number after $1, 2$:
 * $\sigma \left({4}\right) = 7 = 8 - 1$


 * The $2$nd powerful number after $1$


 * The number of faces and vertices of a tetrahedron


 * The number of sides and vertices of a square


 * The number of faces which meet at each vertex of a regular octahedron


 * The number of dimensions in Einstein's space-time


 * The only composite number $n$ such that $n \nmid \left({n - 1}\right)!$: see Divisibility of n-1 Factorial by Composite n‎


 * The smallest $n$ such that $n! + 1$ is square: see Brocard's Problem


 * The $5$th integer $n$ after $0, 1, 2, 3$ such that both $2^n$ and $5^n$ have no zeroes:
 * $2^4 = 16, 5^4 = 625$


 * The $4$th (strictly) positive integer after $1, 2, 3$ which cannot be expressed as the sum of exactly $5$ non-zero squares.


 * The $2$nd even number after $2$ which cannot be expressed as the sum of $2$ composite odd numbers.


 * The $3$rd after $1, 2$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.


 * The only number which equals the number of letters in its name (four) when written in the English language.


 * Every positive integer can be expressed as the sum of at most $4$ squares.


 * The $5$th integer $n$ after $0, 1, 2, 3$ such that $2^n$ contains no zero in its decimal representation:
 * $2^4 = 16$


 * The $2$nd even integer after $2$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes.


 * The $1$st positive integer $n$ such that $n - 2^k$ is prime for all $k$.


 * The $1$st (with $121$) of the $2$ square numbers which are $4$ less than a cube:
 * $4 = 2^2 = 2^3 - 4$


 * The $4$th integer after $0, 1, 2$ which is palindromic in both decimal and ternary:
 * $4_{10} = 11_3$


 * The $3$rd of the $5$ known powers of $2$ whose digits are also all powers of $2$:
 * $1, 2, 4, \ldots$


 * The $5$th number after $0, 1, 2, 3$ which is (trivially) the sum of the increasing powers of its digits taken in order:
 * $4^1 = 4$


 * The $4$th of the trivial $1$-digit pluperfect digital invariants after $1, 2, 3$:
 * $4^1 = 4$


 * The $3$rd positive integer after $2, 3$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $2$nd power of $2$ after $1$ which is the sum of distinct powers of $3$:
 * $4 = 2^2 = 3^0 + 3^1$


 * The number of primes with no more than $1$ digit:
 * $2, 3, 5, 7$

Also see

 * Hyperbola can be Drawn through Four Non-Collinear Points
 * Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts
 * Lagrange's Four Square Theorem, also represented as Hilbert-Waring Theorem for $2$nd Powers
 * Ferrari's Method
 * Four Color Theorem
 * Squares which are 4 Less than Cubes