5

Mathematics

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 ) 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.

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.

Algebra

:Only twelve integers up to 33 cannot be expressed as the sum of five non-zero squares: {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33} where 2, 3 and 7 are the only such primes without an expression. 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.[[File:SporadicGroups.png|thumb|300px|This diagram shows the [[subquotient]] relations of the twenty-six sporadic groups; the five [[Mathieu group]]s form the simplest class (colored red [[File:EllipseSubqR.svg]]). ]]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

Evolution of the Arabic digit

While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in [[File:Text figures 256.svg|45px]].

On the seven-segment display of a calculator and digital clock, it is often represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number. This makes it often indistinguishable from the letter S. Higher segment displays may sometimes may make use of a diagonal for one of the two.

Other fields

In Basque, bost, "5", also means "a lot".

Religion

Judaism

Five is according to Maharal of Prague the number defined as the center point which unifies four extremes.

Islam

The Five Pillars of Islam. The five-pointed simple star ☆ is one of the five used in Islamic Girih tiles.

References

References

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2. {{Cite OEIS
3. {{Cite OEIS. A005385. Safe primes p: (p-1)/2 is also prime
4. {{Cite OEIS. A028388. Good primes
5. {{Cite OEIS. A023201. Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)
6. {{Cite OEIS. A019434. Fermat primes
7. {{Cite OEIS. A007540. Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).
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