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] As no smaller number has more than 12 divisors, 72 is a largely composite number.[6] 72 has an Euler totient of 24.[7] It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72.[8] 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.[8] 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.[9] It also is a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).[10]
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),[11] as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19).[12]
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.[13]
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.[19]
72 is the number of distinct {7/2} magic heptagrams, all with a magic constant of 30.[20]
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).[21]
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 (approximated to the nearest integer).[citation needed]
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.[27][28][29]
The god Osiris was enclosed in a coffin by 72 evil disciples and accomplices of Set.[30]
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).[31][32]
^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.