Four is the smallest composite number, its proper divisors being 1 and 2.
4 is the smallest squared prime (p2) and the only even number in this form. 4 is also the only square one more than a prime number.
A number is a multiple of 4 if its last two digits are a multiple of 4. For example, 1092 is a multiple of 4 because 92 = 4 × 23.
In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, and so on, for any number of up arrows. (That is, 2 [n] 2 = 4 for every positive integer n, where a [n] b is the hyperoperation.)
A four-sided plane figure is a quadrilateral (quadrangle) which include kites, rhombi, rectangles and squares, sometimes also called a tetragon. A circle divided by 4 makes right angles and four quadrants. Because of it, four (4) is the base number of the plane (mathematics). Four cardinal directions, four seasons, the duodecimal system, and the vigesimal system are based on four.
A solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces (as well as vertices) of a polyhedron. The regular tetrahedron is the simplest Platonic solid. A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.
Four-dimensional space is the highest-dimensional space featuring more than three convex regular figures:
Four-dimensional differential manifolds have some unique properties. There is only one differential structure on ℝn except when n = 4, in which case there are uncountably many.
The smallest non-cyclic group has four elements; it is the Klein four-group. Four is also the order of the smallest non-trivial groups that are not simple.
Four is the only integer n for which the (non trivial) alternating group An is not simple.
Four is the maximum number of dimensions of a real associative division algebra (the quaternions), by a theorem of Ferdinand Georg Frobenius.
The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices.
Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.
Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2 − z2.
Four is the highest degree general polynomial equation for which there is a solution in radicals.
The four fours game, there are known solutions for all integers from 0 to 880 (but not 881).