27 (number)

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← 26 27 28 →
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.

Contents

Mathematics

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

The Clebsch surface, with 27 straight lines Clebsch diagonal cubic surface.png
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 . [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 [6]

There are twenty-seven sporadic groups, if the non-strict group of Lie type (with an irreducible representation that is twice that of in 104 dimensions) [7] is included. [8]

In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold for values where is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger [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.

In other fields

See also

Notes

    Related Research Articles

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    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.

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