Mathematics
Integer properties
Twenty is a pronic number, as it is the product of consecutive integers, namely 4 and 5. [3] It is also the second pronic sum number (or pronic pyramid) after 2, being the sum of the first three pronic numbers: 2 + 6 + 12. It is the third composite number to be the product of a squared prime and a prime (and also the second member of the 22 × q family in this form). It is a largely composite number, [4] as it has 6 divisors and no smaller number has more than 6 divisors. It has an aliquot sum of 22; a semiprime, within an aliquot sequence of four composite numbers (20, 22, 14, 10, 8) that belong to the prime 7-aliquot tree. It is the smallest primitive abundant number, [5] and the first number to have an abundance of 2, followed by 104. [6] 20 is the length of a side of the fifth smallest right triangle that forms a primitive Pythagorean triple (20, 21, 29). [7] [lower-alpha 1] It is the third tetrahedral number. [8] In combinatorics, 20 is the number of distinct combinations of 6 items taken 3 at a time. Equivalently, it is the central binomial coefficient for n=3 (sequence A000984 in the OEIS ).
In decimal, 20 is the smallest non-trivial neon number equal to the sum of its digits when raised to the thirteenth power (2013 = 8192 × 1013).[ citation needed ]
Almost integers
Gelfond's constant and pi very nearly have a difference equal to twenty:
Geometric properties
Tessellations
There are twenty edge-to-edge 2-uniform tilings by convex regular polygons, which are uniform tessellations of the plane containing 2 orbits of vertices. [11] [12] 20 is the number of parallelogram polyominoes with 5 cells. [13]
Bring's curve is a Riemann surface of genus four, whose fundamental polygon is a regular hyperbolic twenty-sided icosagon, with an area equal to by the Gauss-Bonnet theorem. [14]
Polyhedra
The largest number of faces a Platonic solid can have is twenty faces, which make up a regular icosahedron. [15] A dodecahedron, on the other hand, has twenty vertices, likewise the most a regular polyhedron can have. [16] There are a total of 20 regular and semiregular polyhedra, aside from the infinite family of semiregular prisms and antiprisms that exists in the third dimension: the 5 Platonic solids, and 15 Archimedean solids (including chiral forms of the snub cube and snub dodecahedron). There are also four uniform compound polyhedra that contain twenty polyhedra (UC13, UC14, UC19, UC33), which is the most any such solids can have; while another twenty uniform compounds contain five polyhedra (that are not part of classes of infinite families, where there exist three more). The compound of twenty octahedra can be obtained by orienting two pairs of compounds of ten octahedra, which can also coincide to yield a regular compound of five octahedra.
Higher-dimensional polytopes
In total, there are 20 semiregular polytopes that only exist up through the 8th dimension, which include 13 Archimedean solids and 7 Gosset polytopes (without counting enantiomorphs, or semiregular prisms and antiprisms).
Abstract algebra
The Happy Family of sporadic groups is made up of twenty finite simple groups that are all subquotients of the friendly giant, the largest of twenty-six sporadic groups. The largest supersingular prime factor that divides the order of the friendly giant is 71, which is the 20th indexed prime number, where 26 also represents the number of partitions of 20 into prime parts. [17] Both 71 and 20 represent self-convolved Fibonacci numbers, respectively the seventh and fifth members in this sequence . [18] [19]
Rubik's cube
20 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. [20] [21]
