4

4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures.

Four is the smallest composite number, its proper divisors being 1 and 2.^[1] Four is the sum and product of two with itself: , the only number b such that , which also makes four the smallest squared prime number. In Knuth's up-arrow notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, and so forth, for any number of up arrows.^[2] By consequence, four is the only square one more than a prime number, specifically three. The sum of the first four prime numbers two + three + ${\displaystyle 2+2=4=2\times 2}$${\displaystyle a+a=b=a\times a}$ ${\displaystyle 2^{2}}$five + seven is the only sum of four consecutive prime numbers that yields an odd prime number, seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, three and five, which are the first two Fermat primes, like seventeen, which is the third. On the other hand, the square of four (), equivalently the fourth power of two (), is sixteen; the only number that has as a form of factorization${\displaystyle 4^{2}}$${\displaystyle 2^{4}}$${\displaystyle a^{b}=b^{a}}$. Holistically, there are four elementary arithmetic operations in mathematics: addition (+), subtraction (−), multiplication (×), and division (÷); and four basic number systems, the real numbers , rational numbers, integers, and natural numbers.${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {N} }$

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. . A number is a multiple of 4 if its last two digits are a multiple of 4.^[3] For example, 1092 is a multiple of 4 because 92 = 4 × 23.${\displaystyle 4x=y^{2}-z^{2}}$

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers.^[4] Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.^[5]

There are four all-Harshad numbers: 1, 2, 4, and 6. 12, which is divisible by four thrice over, is a Harshad number in all bases except octal.

A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon. It can be further classified as a rectangle or oblong, kite, rhombus, and square.