27 (number)

← 26 27 28 →
← 20 21 22 23 24 25 26 27 28 29 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-seven
Ordinal	27th
Factorization	33
Divisors	1, 3, 9, 27
Greek numeral	ΚΖ´
Roman numeral	XXVII, xxvii
Binary	110112
Ternary	10003
Senary	436
Octal	338
Duodecimal	2312
Hexadecimal	1B16

27 (twenty-seven) is the natural number following 26 and preceding 28.

Mathematics

Including the null-motif, there are 27 distinct hypergraph motifs. ^[1]

The Clebsch surface, with 27 straight lines

There are exactly twenty-seven straight lines on a smooth cubic surface, ^[2] which give a basis of the fundamental representation of Lie algebra ${\displaystyle \mathrm {E_{6}} }$. ^{[3] [4]}

The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional; ^[5] its automorphism group is the 52-dimensional exceptional Lie algebra ${\displaystyle \mathrm {F_{4}} .}$ ^[6]

There are twenty-seven sporadic groups, if the non-strict group of Lie type ${\displaystyle \mathrm {T} }$ (with an irreducible representation that is twice that of ${\displaystyle \mathrm {F_{4}} }$ in 104 dimensions) ^[7] is included. ^[8]

In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold ${\displaystyle \sigma (n)<e^{\gamma }n\log \log n}$ for values ${\displaystyle n\leq 5040,}$ where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger ${\displaystyle n.}$ ^{[9] [10] [11]}

The Clebsch surface has 27 exceptional lines can be defined over the real numbers.

It is possible to arrange 27 vertices and connect them with edges to create the Holt graph.

27 is 3³, and therefore, it is the second tetration of 3 (²3).

In other fields

• The 27 club refers to the age when many popular music figures died. It connects the age 27 with connotations of an early end.^{[ citation needed ]}

See also

• 72 (number) – 27 reversed

References

1. Lee, Geon; Ko, Jihoon; Shin, Kijung (2020). "Hypergraph Motifs: Concepts, Algorithms, and Discoveries". In Balazinska, Magdalena; Zhou, Xiaofang (eds.). 46th International Conference on Very Large Data Bases. Proceedings of the VLDB Endowment. Vol. 13. ACM Digital Library. pp. 2256–2269. arXiv: 2003.01853 . doi:10.14778/3407790.3407823. ISBN   9781713816126. OCLC   1246551346. S2CID   221779386.
2. Baez, John Carlos (February 15, 2016). "27 Lines on a Cubic Surface". AMS Blogs. American Mathematical Society . Retrieved October 31, 2023.
3. Aschbacher, Michael (1987). "The 27-dimensional module for E₆. I". Inventiones Mathematicae . 89. Heidelberg, DE: Springer: 166–172. Bibcode:1987InMat..89..159A. doi:10.1007/BF01404676. MR   0892190. S2CID   121262085. Zbl   0629.20018.
4. Sloane, N. J. A. (ed.). "SequenceA121737(Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved October 31, 2023.
5. Kac, Victor Grigorievich (1977). "Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras". Communications in Algebra . 5 (13). Taylor & Francis: 1380. doi:10.1080/00927877708822224. MR   0498755. S2CID   122274196. Zbl   0367.17007.
6. Baez, John Carlos (2002). "The Octonions". Bulletin of the American Mathematical Society . 39 (2). Providence, RI: American Mathematical Society: 189–191. doi: 10.1090/S0273-0979-01-00934-X . MR   1886087. S2CID   586512. Zbl   1026.17001.
7. Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type" . Communications in Algebra . 29 (5). Philadelphia, PA: Taylor & Francis: 2151. doi:10.1081/AGB-100002175. MR   1837968. S2CID   122060727. Zbl   1004.20003.
8. Hartley, Michael I.; Hulpke, Alexander (2010). "Polytopes Derived from Sporadic Simple Groups". Contributions to Discrete Mathematics. 5 (2). Alberta, CA: University of Calgary Department of Mathematics and Statistics: 27. doi: 10.11575/cdm.v5i2.61945 . ISSN   1715-0868. MR   2791293. S2CID   40845205. Zbl   1320.51021.
9. Axler, Christian (2023). "On Robin's inequality". The Ramanujan Journal . 61 (3). Heidelberg, GE: Springer: 909–919. arXiv: 2110.13478 . Bibcode:2021arXiv211013478A. doi: 10.1007/s11139-022-00683-0 . S2CID   239885788. Zbl   1532.11010.
10. Robin, Guy (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann" (PDF). Journal de Mathématiques Pures et Appliquées . Neuvième Série (in French). 63 (2): 187–213. ISSN   0021-7824. MR   0774171. Zbl   0516.10036.
11. Sloane, N. J. A. (ed.). "SequenceA067698(Positive integers such that sigma(n) is greater than or equal to exp(gamma) * n * log(log(n)).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved October 31, 2023.

Further reading

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106.

External links

• Prime Curios! 27 from the Prime Pages
• Mystery of the number 27 - Large collection of 27 related trivia and facts.
• The 27 Project - collection of sightings of 27 in movies, TV, culture and art