Number 4 in Math

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The smallest composite number, the smallest square number: 22.

The number of prime numbers less than 10: 2, 3, 5 and 7.

The number of sides/angles of a square, a rectangular, a rhombus, a parallelogram and a trapezoid.

The number of equilateral triangular faces and of vertices of a pyramid (or tetrahedron, 4 vertices/6 edges/4 equilateral triangle faces).

The smallest number of faces (or vertices) of a regular polyhedron. It is of a pyramid (or tetrahedron).

The number of elementary arithmetic operations: addition (+), subtraction (–), multiplication (´) and division (¸).

A number is divisible by 4 if and only if its last 2 digits also form a number divisible by 4.

There are only 4 triangular numbers that are also Fibonacci numbers: 1, 3, 21 and 55.

The 4th Fibonacci number is the only Fibonacci number that is a prime number but its subscript is composite.

The 3rd Lucas number.

The number of colors needed to paint any planar map such that no 2 adjacent countries (sharing a non-zero length border) have the same color. In 1976, Wolfgang Haken and Kenneth Appel’s “Every planar map is four colorable” claimed to prove this four-color conjecture (by Francis Guthrie, 1852), but the computer-aid proof was too complicated to check.

The highest degree for which an algebraic polynomial equation is always solvable by radicals, (i.e. in terms of additions, subtractions, multiplications, divisions and root extractions).

The smallest order of a finite field that is not a prime finite field.

(3, 4, 5) is the smallest a Pythagorean triple (lengths of 3 sides of a right triangle).

There is exactly one sphere passing through any 4 non-coplanar points in the 3-dimension Euclidean space.

Each ratio 15768/3942 = 17568/4392 = 23184/5796 = 31824/7956 = 4, using each of 9 digits 1-9 once, shows how to arrange a 9-book set on 2 shelves to mark the book #4.

6952 = 4×1738, using each of 9 digits 1-9 once.

852 = 4×1963, using each of 9 digits 1-9 once.

The equality 7039×4 = 28,156 uses each of 10 digits 0-9 once.

85,427,136 = 2×42,713,568 = 4×21,356,784 = 6×14,237,856, using 3 permutations of the number itself (no digit 9 in all numbers).

076,923×4 = 307,692.

047,619×4 = 190,476.

190,476×4 = 761,904.

095,238×4 = 380,952.

142,857×4 =571,428.

238,095×4 = 952,380.

102,564×4 = 410,256.

025,641×4 = 102,564.

153,846×4 = 615,384.

128,205×4 = 512,820.

179,487×4 = 717,948.

205,128×4 = 820,512.

230,769×4 = 923,076.

The number of groups of conic curves: circle, ellipse, parabola and hyperbola, formed by the intersection of a right circular cone and a plane.

In a triangle, the Euler line goes through 4 center points: the circumcenter, the centroid, the orthocenter and the center of the Euler’s 9-point circle.

Its odd powers end with 4. Its even powers end with 6.

48 = 82–42 = (4+8)×4, where 8 = 4×2.

Sum of the first 3 factorials, including 0!: 0!+1!+2! = 4.

Two fourth powers using the same digits: 44 = 256 and 54 = 625.

8,712 = 2,178×4 and 8,712×2,178 = 18,974,736 = 664.

Every integer can be expresses as a sum of 4 squares.

Cousin prime numbers are two prime numbers whose difference is 4.

A well-know, even complicated, formula:

arctan(1/239) = 4×arctan(1/5) – pi/4 = arctan(1/70) – arctan(1/99) =

= arctan(1/408) + arctan(1/577).

The smallest order for which there exist 2 non-isomorphic abelian finite groups (Z2´Z2 and Z4).

(*)   8712 = 2178×4 & 8712×2178 = 18,974,736 = 43562 (= 664).

87128712 = 21782178×4 & 87128712×21782178 = 435643562

(**)   87912 = 21978×4 & 21978×87912 = 1,932,129,936 = 439562

8791287912 = 21978…21978×4 & 21978…21978×87912…87912 = 43956…43956

(***)   879912 = 219978×4 & 219978×879912 = 193,561,281,936 = 4399562

879912879912 = 219978219978×4 &

219978219978×879912879912 = 4399564399562

(****)   879…912 = 219…978×4 & 219…978×879…912 = 439…9562

879…912…879…912 = 219…978…219…978×4 &

219...978...219...978×879...912...879...912 = 439...956...439...956

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