72 (number)

In mathematics

72 is a pronic number, as it is the product of 8 and 9. It is the smallest Achilles number, as it is a powerful number that is not itself a power.

72 is an abundant number. With exactly twelve positive divisors, including 12 (one of only two sublime numbers), 72 is also the twelfth member in the sequence of refactorable numbers. :The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ... As no smaller number has more than 12 divisors, 72 is a largely composite number. 72 has an Euler totient of 24. It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72. It is equal to the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors. 144, or twice 72, is also highly totient, as is 576, the square of 24. While 17 different integers have a totient value of 72, the sum of Euler's totient function φ(x) over the first 15 integers is 72. It is also a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).

| 72 is the second multiple of 12, after 48, that is not a sum of twin primes.

It is, however, the sum of four consecutive primes , as well as the sum of six consecutive primes . | 72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways: . |72 is the sum of the first two sphenic numbers (30, 42), which have a difference of 12, that is also their abundance. | 72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 195 + 435 + 465 + 475 + 675 = 725. | 72 is the magic constant of the first non-normal, full prime reciprocal magic square in decimal, based on in a 16 × 16 grid. | 72 is the sum between 60 and 12, the former being the second unitary perfect number before 6 (and the latter the smallest of only two sublime numbers).

More specifically, twelve is also the number of divisors of 60, as the smallest number with this many divisors. | 72 is the number of distinct {7/2} magic heptagrams, all with a magic constant of 30. | 72 is the sum of the eighth row of Lozanić's triangle, and equal to the sum of the previous four rows (36, 20, 10, 6).

As such, this row is the third and largest to be in equivalence with a sum of consecutive row sums, after (1, 2, 3; 6) and (6, 10, 20; 36). | 72 is the number of degrees in the central angle of a regular pentagon, which is constructible with a compass and straight-edge.

72 plays a role in the rule of 72 in economics when approximating annual compounding of interest rates of a round 6% to 10%, due in part to its high number of divisors.

Inside \mathrm E_{n} Lie algebras:

• 72 is the number of vertices of the six-dimensional 122 polytope, which also contains as facets 720 edges, 702 polychoral 4-faces, of which 270 are four-dimensional 16-cells, and two sets of 27 demipenteract 5-faces. These 72 vertices are the root vectors of the simple Lie group \mathrm E_{6}, which as a honeycomb under 222 forms the \mathrm E_{6} lattice. 122 is part of a family of k22 polytopes whose first member is the fourth-dimensional 3-3 duoprism, of symmetry order 72 and made of six triangular prisms. On the other hand, 321 ∈ k21 is the only semiregular polytope in the seventh dimension, also featuring a total of 702 6-faces of which 576 are 6-simplexes and 126 are 6-orthoplexes that contain 60 edges and 12 vertices, or collectively 72 one-dimensional and two-dimensional elements; with 126 the number of root vectors in \mathrm E_{7}, which are contained in the vertices of 231 ∈ k31, also with 576 or 242 6-simplexes like 321. The triangular prism is the root polytope in the k21 family of polytopes, which is the simplest semiregular polytope, with k31 rooted in the analogous four-dimensional tetrahedral prism that has four triangular prisms alongside two tetrahedra as cells.
• The complex Hessian polyhedron in \mathbb{C}^3 contains 72 regular complex triangular edges, as well as 27 polygonal Möbius–Kantor faces and 27 vertices. It is notable for being the vertex figure of the complex Witting polytope, which shares 240 vertices with the eight-dimensional semiregular 421 polytope whose vertices in turn represent the root vectors of the simple Lie group \mathrm E_{8}.

There are 72 compact and paracompact Coxeter groups of ranks four through ten: 14 of these are compact finite representations in only three-dimensional and four-dimensional spaces, with the remaining 58 paracompact or noncompact infinite representations in dimensions three through nine. These terminate with three paracompact groups in the ninth dimension, of which the most important is \tilde {T}{9}: it contains the final semiregular hyperbolic honeycomb 621 made of only regular facets and the 521 Euclidean honeycomb as its vertex figure, which is the geometric representation of the \mathrm E{8} lattice. Furthermore, \tilde {T}{9} shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form \mathrm E{8}++ equivalent to the tenth-dimensional symmetries of Lie algebra \mathrm E_{10}.

72 lies between the 8th pair of twin primes (71, 73), where 71 is the largest supersingular prime that is a factor of the largest sporadic group (the friendly giant \mathbb {F_{1}}), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers :{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73} that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of \mathbb {F_{1}}. Sporadic groups are a family of twenty-six finite simple groups, where \mathrm E_{6}, \mathrm E_{7}, and \mathrm E_{8} are associated exceptional groups that are part of sixteen finite Lie groups that are also simple, or non-trivial groups whose only normal subgroups are the trivial group and the groups themselves.

In religion

• In Islam, 72 is the number of beautiful wives that are promised to martyrs in paradise, according to Hadith (sayings of Muhammad).

In other fields

72 is also:

• In typography, a point is 1/72 inch.
• The rule of 72 in finance.
• 72 equal temperament is a tuning used in Byzantine music and by some modern composers.
• The number of micro seasons in the traditional Japanese calendar

Notes

References

References

1. {{Cite OEIS. A002378. Oblong (or promic, pronic, or heteromecic) numbers
2. {{Cite OEIS. A052486. Achilles numbers - powerful but imperfect.
3. {{Cite OEIS. A005101. Abundant numbers (sum of divisors of m exceeds 2m).
4. {{Cite OEIS. A081357. Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.
5. {{Cite OEIS. A067128. Ramanujan's largely composite numbers
6. {{Cite OEIS. A000010. Euler totient function.
7. {{Cite OEIS. A097942. Highly totient numbers.
8. {{Cite OEIS. A002088. Sum of totient function.
9. {{Cite OEIS. A005349. Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.
10. {{Cite OEIS. A034963. Sums of four consecutive primes.
11. {{Cite OEIS. A127333. Numbers that are the sum of 6 consecutive primes.
12. {{Cite OEIS. A090788. Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.
13. {{Cite OEIS. A007304. Sphenic numbers: products of 3 distinct primes.
14. {{Cite OEIS. A005101. Abundant numbers (sum of divisors of m exceeds 2m).
15. {{Cite OEIS. A033880. Abundance of n, or (sum of divisors of n) - 2n.
16. David Wells: The Penguin Dictionary of Curious and Interesting Numbers
17. Subramani, K.. (2020). "On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p - 1.". S.M.A.R.T..
18. {{Cite OEIS. A007450. Decimal expansion of 1/17.
19. {{Cite OEIS. A005179. Smallest number with exactly n divisors.
20. {{Cite OEIS. A200720. Number of distinct normal magic stars of type {n/2}.
21. {{Cite OEIS. A005418. ...row sums of Losanitsch's triangle.
22. (2015). "Sporadic and Exceptional".
23. Jami`at-Tirmidhi. "The Book on Virtues of Jihad, Vol. 3, Book 20, Hadith 1663".
24. (2009). "Fully Committed: Suicide Bombers' Motivation and the Quest for Personal Significance". Political Psychology.
25. W3C. "CSS Units".
26. (16 October 2015). "Japan's 72 Microseasons".