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72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen (i.e., 60 in duodecimal).

In mathematics

Seventy-two is a pronic number, as it is the product of 8 and 9. It is the smallest Achilles number, as it's 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. 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 also is 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 (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19).
• 72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways: 11 − 7 = 19 − 17.
• 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 magic constant of the first non-normal, full prime reciprocal magic square in decimal, based on ⁠1/17⁠ 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 k 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 ${\displaystyle \mathrm {E} _{n}}$ 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 ${\displaystyle \mathrm {E} _{6}}$, which as a honeycomb under 2₂₂ forms the ${\displaystyle \mathrm {E} _{6}}$ 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 ${\displaystyle \mathrm {E} _{7}}$, which are contained in the vertices of 2₃₁ ∈ k₃₁, also with 576 or 24 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 ${\displaystyle \mathbb {C} ^{3}}$ 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 ${\displaystyle \mathrm {E} _{8}}$.

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 ${\displaystyle {\tilde {T}}_{9}}$: 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 ${\displaystyle \mathrm {E} _{8}}$ lattice. Furthermore, ${\displaystyle {\tilde {T}}_{9}}$ shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form ${\displaystyle \mathrm {E} _{8}}$ equivalent to the tenth-dimensional symmetries of Lie algebra ${\displaystyle \mathrm {E} _{10}}$.

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 ${\displaystyle \mathbb {F_{1}} }$), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of ${\displaystyle \mathbb {F_{1}} }$. Sporadic groups are a family of twenty-six finite simple groups, where ${\displaystyle \mathrm {E} _{6}}$, ${\displaystyle \mathrm {E} _{7}}$, and ${\displaystyle \mathrm {E} _{8}}$ 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 science

In chemistry

• The atomic number of hafnium

In astronomy

• Messier object M72, a magnitude 10.0 globular cluster in the constellation Aquarius.
• The New General Catalogue object NGC 72, a magnitude 13.5 barred spiral galaxy in the constellation Andromeda.
• The precession of equinoxes traces out a pair of cones joined at their apices in a cycle of approximately 26,000 years, that is 1 degree every 72 years (approximated to the nearest integer).

In religion

• The number of languages spoken at the Tower of Babylon, according to later tradition.
• The conventional number of scholars translating the Septuagint, according to the legendary account in the "Letter of Aristeas".
• The number of companions of Zoroaster who were killed
• The conventional number of disciples sent forth by Jesus in Luke 10 in some manuscripts (seventy in others).
• The number of names of God, according to Kabbalah (see names of God in Judaism).
• The Shemhamphorasch related to the number of the names of God.
• The total number of books in the Bible in the Catholic version if the Book of Lamentations is considered part of the Book of Jeremiah.
• The current distribution of the Book of Revelation is 22 chapters, adopted since the 13th century, but the oldest known division of the text is that of the Greek commentator Andrew of Cesary (6th century) in 72 chapters.
• The number of people killed along with Imam Hussain at the Battle of Karbala.
• The degrees of the Jacob's Ladder were to the number of 72, according to the Zohar.
• The 72 disciples of Confucius who mastered his teachings (also given as 77).
• Mahavira, the twenty-fourth and last tirthankara of Jainism, is said to have attained nirvana after his physical death at the age of 72.
• Thoth, in an Egyptian creation myth, wins a 72nd of each day of the year from the Moon in a game of draughts, as a favour to Nut, the Sky Goddess. He uses these portions to make the five intercalary days on which the remaining Gods and Goddesses are born.
• The god Osiris was enclosed in a coffin by 72 evil disciples and accomplices of Set.
• At the age of the puberty, the young Parsee received the investiture of the sacred cord Kucti made of 72 linens in symbol of the community.
• In Cao Đài, the number of planets between hell and heaven.
• There are 72 stupas which comprise Borobudur, the world's largest Buddhist temple.
• 72 major temples have been found at Angkor, seat of the ancient Khmer Empire.
• In Islam, 72 is the number of sects or denominations that are doomed to Hell, according to Hadith (sayings of Muhammad).
• The number of demons sealed away by King Solomon with The Lesser Key of Solomon.
• 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

Seventy-two is also:

• In typography, a point is 1/72 inch.
• The number of the French department Sarthe.*The number of the French department Sarthe.
• The registry of the U.S. Navy's nuclear aircraft carrier USS Abraham Lincoln (CVN-72), named after U.S. President Abraham Lincoln.
• The designation of the Soviet T-72 tank.
• The Rule of 72 in finance.
• Book: 72 Hour Hold by Bebe Moore Campbell.
• CD: Seventy Two & Sunny by Uncle Kracker.
• Jill Clayburgh and LeVar Burton starred in Firestorm: 72 Hours in Oakland (1993).
• Alternative music band The Delta 72.
• The Persian classical santur, a hammered dulcimer, has 72 strings in 24 triple-stringed courses.
• The Turin Brakes song, also known as Emergency 72
• The number of members in National Senate of Argentina.
• A Civil Air Patrol unit in Laramie, WY, RMR-WY-072.
• There are 72 demons and other spirits in the goetia The Lesser Key of Solomon.
• A common limit for characters per line in computing.
• 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

In sports and games

• The usual par for an 18-hole golf course, especially those in tournament play.
• The number of spaces in a game of Parcheesi, from start space to "home".

Notes

1. Where 71 is also the largest prime number less than 73 that is not a member of this set.
2. The only other finite simple groups are the infinite families of cyclic groups and alternating groups. An exception is the Tits group ${\displaystyle \mathbb {T} }$, which is sometimes considered a 17th non-strict group of Lie type that can otherwise more loosely classify as a 27th sporadic group.

References

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   The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ...

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    {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}

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External links

• Go Figure: What can 72 tell us about life, BBC News, 20 July 2011

• Integers