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

Mathematics

Including the null-motif, there are 27 distinct hypergraph motifs.

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

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

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) is included.

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 ${\displaystyle n.}$

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.

See also

• 72 (number) – 27 reversed

Notes

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.). "Sequence A121737 (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.). "Sequence A067698 (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

• Integers