5 is a
Fermat prime, a
Mersenne prime exponent, as well as a
Fibonacci number. 5 is the first
congruent number, as well as the length of the
hypotenuse of the smallest integer-sided
right triangle, making part of the smallest
Pythagorean triple (
3,
4, 5). 5 is the first
safe prime and the first
good prime. 11 forms the first pair of
sexy primes with 5. 5 is the second
Fermat prime, of a total of five known Fermat primes. 5 is also the first of three known
Wilson primes (5, 13, 563).
Geometry A shape with five sides is called a
pentagon. The equilateral pentagon is the first
regular polygon that does not
tile the
plane with copies of itself. The pentagon solid has the largest
face of any of the five regular three-dimensional regular
Platonic solids. A
conic is
determined using five points in the same way that two points are needed to determine a
line. A
pentagram, or five-pointed
polygram, is a
star polygon constructed by connecting some non-adjacent vertices of a
regular pentagon as
self-intersecting edges. The internal geometry of the pentagon and pentagram (represented by its
Schläfli symbol {{math|1={5/2} }}) appears prominently in
Penrose tilings. Pentagrams are
facets inside
Kepler–Poinsot star polyhedra and
Schläfli–Hess star polychora. There are five regular
Platonic solids the
tetrahedron, the
cube, the
octahedron, the
dodecahedron, and the
icosahedron. The plane contains a total of five
Bravais lattices, or arrays of
points defined by discrete
translation operations.
Uniform tilings of the plane, are generated from combinations of only five regular polygons. Five-fold symmetry is associated with the
golden ratio \frac{1+\sqrt 5}{2}. In a regular
pentagram, intersections of diagonals divide one another in the golden ratio.
Higher dimensional geometry A
hypertetrahedron, or 5-cell, is the 4 dimensional analogue of the
tetrahedron. It has five vertices. Its orthographic projection is
homomorphic to the group
K5. There are five fundamental
mirror symmetry point group families in 4-dimensions. There are also 5
compact hyperbolic Coxeter groups, or
4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams. is the simplest regular
polychoron.
Arithmetic 5 is the value of the central
cell of the first non-trivial
normal magic square, called the
Luoshu square. All
integers n \ge 34 can be expressed as the sum of five non-zero
squares. There are five countably infinite
Ramsey classes of
permutations. 5 is
conjectured to be the only
odd,
untouchable number; if this is the case, then five will be the only odd prime number that is not the base of an
aliquot tree. relations of the twenty-six
sporadic groups; the five
Mathieu groups form the simplest class (colored red ). Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers;
Helfgott has provided a proof of this (also known as the
odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes
peer-review. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).
Group theory In
graph theory, all
graphs with four or fewer vertices are
planar, however, there is a graph with five vertices that is not:
K5, the
complete graph with five vertices. By
Kuratowski's theorem, a finite graph is planar
if and only if it does not contain a subgraph that is a subdivision of
K5, or
K3,3, the
utility graph. There are five complex
exceptional Lie algebras. The five
Mathieu groups constitute the
first generation in the
happy family of
sporadic groups. These are also the first five sporadic groups
to have been described. A
centralizer of an element of order 5 inside the
largest sporadic group \mathrm {F_1} arises from the product between
Harada–Norton sporadic group \mathrm{HN} and a group of order 5. == List of basic calculations ==