72 is an
abundant number.[3] With exactly twelve positive divisors, including
12 (one of only two
sublime numbers),[4] 72 is also the twelfth member in the sequence of
refactorable numbers.[5] 72 has a
Euler totient of
24,[6] which makes it a
highly totient number, as there are
17 solutions to the equation φ(x) = 72, more than any integer under 72.[7] 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.[7] 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.[8] It also is a perfect indexed
Harshad number in
decimal (twenty-eighth), as it is divisible by the sum of its digits (
9).[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),[10] as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19).[11]
72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways: 112 − 72 = 192 − 172.[12]
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.[18]
72 is the number of distinct {7/2} magic heptagrams, all with a magic constant of 30.[19]
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).[20]
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).
There are 72
compact and
paracompactCoxeter 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 : it contains the final
semiregular hyperbolic honeycomb
621 made of only
regular facets and the
521Euclidean honeycomb as its
vertex figure, which is the geometric representation of the
lattice. Furthermore, shares the same fundamental symmetries with the Coxeter-Dynkin
over-extended form ++ equivalent to the
tenth-dimensional symmetries of Lie algebra .
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 (approximation to the nearmost 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[24]
The conventional number of
disciples sent forth by Jesus in Luke 10 in some manuscripts (seventy in others).
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 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.[25][26][27]
The god
Osiris was enclosed in a coffin by 72 evil disciples and accomplices of
Set.[28]
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).[29][30]
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
^Where 71 is also the largest prime number less than 73 that is not a member of this set.
^The only other finite simple groups are the infinite families of
cyclic groups and
alternating groups. An exception is the
Tits group, which is sometimes considered a 17th non-strict group of Lie type that can otherwise more loosely classify as a 27th sporadic group.