# 72 (number)

 ← 71 72 73 →
Cardinalseventy-two
Ordinal72nd
(seventy-second)
Factorization23 × 32
Divisors1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Greek numeralΟΒ´
Roman numeralLXXII
Binary10010002
Ternary22003
Senary2006
Octal1108
Duodecimal6012

72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen (i.e., 60 in duodecimal).

## In mathematics

Seventy-two is a pronic number, as it is the product of 8 and 9.[1] It is the smallest Achilles number, as it's a powerful number that is not itself a power.[2]

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 of the first two sphenic numbers (30, 42),[13] which have a difference of 12, that is also their abundance.[14][15]
• 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.[16][17]
• 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).
• 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:

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[22][a] that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of ${\displaystyle \mathbb {F_{1}} }$.[23] 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.[b]

## 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 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.[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 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).[31][32]

## In other fields

Seventy-two is also:

## 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

1. ^ Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
2. ^ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
3. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
4. ^ Sloane, N. J. A. (ed.). "Sequence A081357 (Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
5. ^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ...
6. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
7. ^ a b Sloane, N. J. A. (ed.). "Sequence A097942 (Highly totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
8. ^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
9. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
10. ^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02.
11. ^ Sloane, N. J. A. (ed.). "Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02.
12. ^ Sloane, N. J. A. (ed.). "Sequence A090788 (Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03.
13. ^ Sloane, N. J. A. (ed.). "Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13.
14. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13.
15. ^ Sloane, N. J. A. (ed.). "Sequence A033880 (Abundance of n, or (sum of divisors of n) - 2n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13.
16. ^ Subramani, K. (2020). "On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p - 1" (PDF). J. Of Math. Sci. & Comp. Math. 1 (2). Auburn, WA: S.M.A.R.T.: 198–200. doi:10.15864/jmscm.1204. eISSN 2644-3368. S2CID 235037714.
17. ^ Sloane, N. J. A. (ed.). "Sequence A007450 (Decimal expansion of 1/17.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-24.
18. ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-11.
19. ^ Sloane, N. J. A. (ed.). "Sequence A200720 (Number of distinct normal magic stars of type {n/2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-09.
20. ^ Sloane, N. J. A. (ed.). "Sequence A005418 (...row sums of Losanitsch's triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
21. ^ David Wells: The Penguin Dictionary of Curious and Interesting Numbers
22. ^ Sloane, N. J. A. (ed.). "Sequence A154363 (Numbers from Bhargava's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}
23. ^ He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". p. 20. arXiv:1505.06742 [math.AG].
24. ^ "صدویک نام خداوند چیست؟" (in Persian). Amordad. 22 May 2018. Retrieved 8 January 2023.
25. ^ Hart, George (1990). Egyptian Myths. University of Texas Press. pp. 40–41. ISBN 0-292-72076-9.
26. ^ Hart, George (1990). A Dictionary of Egyptian Gods and Goddesses. University of Texas Press. pp. 144–145. ISBN 9780292720763.
27. ^ Plutarch. Isis and Osiris. Loeb Classics. pp. LCL 306: 30–31.
28. ^ "Egyptian Myths", George Hart, p41, University of Texas Press, 1990
29. ^ "Sects In Islam - 73 Groups in Islam, Division - Denominations". Archived from the original on 6 May 2013. Retrieved 24 March 2013.
30. ^ Sunan Ibn Maajah, no. 3982 "My Ummah will be divided into seventy-three sects, one of which will be in Paradise and seventy-two will be in the Fire"
31. ^ Jami`at-Tirmidhi. "The Book on Virtues of Jihad, Vol. 3, Book 20, Hadith 1663". Sunnah.com - Sayings and Teachings of Prophet Muhammad (صلى الله عليه و سلم). Retrieved 2024-04-02.
32. ^ Kruglanski, Arie W.; Chen, Xiaoyan; Dechesne, Mark; Fishman, Shira; Orehek, Edward (2009). "Fully Committed: Suicide Bombers' Motivation and the Quest for Personal Significance". Political Psychology. 30 (3): 331–357. ISSN 0162-895X.
33. ^ "Japan's 72 Microseasons". 16 October 2015.