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
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

Twenty-seven is the cube of 3, or the 2nd tetration of 3: ²3 = 3³ = 3 × 3 × 3. It is divisible by the number of prime numbers below it (nine).

The first non-trivial decagonal number is 27. ^[1]

27 has an aliquot sum of 13 ^[2] (the sixth prime number) in the aliquot sequence (27, 13, 1, 0) of only one composite number, rooted in the 13-aliquot tree. ^[3]

In the Collatz conjecture (i.e. the ${\displaystyle 3n+1}$ problem), a starting value of 27 requires 3  ×  37  =  111 steps to reach 1, more than any smaller number. ^{[4] [a]}

27 is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and 39 adding to twice 27. ^{[7] [b]}

A prime reciprocal magic square based on multiples of ${\displaystyle {\tfrac {1}{7}}}$ in a 6×6 square has a magic constant of 27.

Including the null-motif, there are 27 distinct hypergraph motifs. ^[8]

The Clebsch surface, with 27 straight lines

There are exactly twenty-seven straight lines on a smooth cubic surface, ^[9] which give a basis of the fundamental representation of Lie algebra ${\displaystyle \mathrm {E_{6}} }$. ^{[10] [11]}

The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional; ^[12] its automorphism group is the 52-dimensional exceptional Lie algebra ${\displaystyle \mathrm {F_{4}} .}$ ^[13]

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) ^[14] is included. ^[15]

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 ${\displaystyle n.}$ ^{[16] [17] [18]}

Base-specific

In decimal, 27 is the first composite number not divisible by any of its digits, as well as:

• the third Smith number ^[19] and sixteenth Harshad number, ^[20]
• the only positive integer that is three times the sum of its digits,
• equal to the sum of the numbers between and including its digits: 2 + 3 + 4 + 5 + 6 + 7 = 27.

Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27.

• In similar fashion, any multiple of 27 can be mirrored and spaced with a zero each for another multiple of 27 (i.e. 27 and 702, 54 and 405, and 378 and 80703 are all multiples of 27).
• Any multiple of 27 with "000" or "999" inserted yields another multiple of 27 (20007, 29997, 50004, and 59994 are all multiples of 27).

In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513.

In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in π:

$${\displaystyle 3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color {red}27}9\ldots }$$

If one starts counting with zero, 27 is the second self-locating string after 6, of only a few known. ^{[21] [22]}

In science

• The Moon revolves around the Earth the same direction as Earth spins but 27 (27.3) times slower.
• 27km is the circumference of the LHC located at CERN in Meyrin Switzerland
• The atomic number of cobalt.
• Dark matter is thought to make up 27% of the universe. ^[23]
• 27 is the number of bones in the human hand. ^[24]

Astronomy

• The Messier object M27, a magnitude 7.5 planetary nebula in the constellation Vulpecula, also known as the Dumbbell Nebula.
• The New General Catalogue object NGC 27, a spiral galaxy in the Andromeda constellation.
• The Saros number of the solar eclipse series, which began on March 9, 1993, BCE and ended on April 16, 713 BCE. ^[25] The duration of Saros series 27 was 1,280.1 years, and it contained 72 solar eclipses. Further, the Saros number of the lunar eclipse series, which began on July 28, 1926 BCE and ended on January 23, 411 BCE. ^[26] The duration of Saros series 27 was 1532.5 years, and it contained 86 lunar eclipses.

Electronics

• The type 27 vacuum tube (valve), a triode introduced in 1927, was the first tube mass-produced for commercial use to incorporate an indirectly heated cathode. This made it the first vacuum tube that could function as a detector in AC-powered radios. Prior to the introduction of the 27, home radios were powered by a set of three or more storage batteries with voltages of 3 to 135 volts.

In language and literature

• The number of letters in the Spanish alphabet. ^[27]
• The number of books in the New Testament.
• The number of generations from David to the Christ according to the Gospel of Matthew. (Mt 1,1-17)
• The number of Grievances of the United States Declaration of Independence.
• The number of ratified Amendments to the United States Constitution.
• Until 1835, the English Alphabet consisted of 27 letters: after "Z" the 27th letter of the alphabet was Ampersand (&)
• The total number of letters in the Hebrew alphabet (22 regular letters and 5 final consonants).
• Alternate name for The Hunt, a book by William Diehl.
• Abbé Faria's prisoner number in the book The Count of Monte Cristo .
• In Stephen King's novel It , It returns every 27 years to Derry.
• The war led by the Antichrist will begin in the year 1999 and will last 27 years according to Nostradamus.

In astrology

• 27 Nakṣatra or lunar mansions in Hindu astrology.

In music

• "27", the name of a song by Biffy Clyro on their 2002 debut album, Blackened Sky
• "27", the name of a song by Title Fight on their 2011 album, Shed

In sports

• The value of all the colors in snooker add up to 27.
• The number of outs in a regulation baseball game for each team at all adult levels, including professional play, is 27.
• The New York Yankees have won 27 World Series championships, the most of any team in the MLB.

In other fields

Twenty-seven is also:

• A-27, American attack aircraft.
• The code for international direct-dial phone calls to South Africa.
• The name of a cigarette, Marlboro Blend No. 27.
• The number of the French department Eure.
• The number of the Chiefs and Elders in the Maghan Gbara, the traditional authority of a tribal confederation in Ghana.
• The current number of countries in the European Union, as of 2024.

See also

• List of highways numbered 27
• List of highways numbered 27A

Notes

1. On the other hand,
   • The reduced Collatz sequence of 27, that counts the number of prime numbers in its trajectory, is 41. ^[5]
     This count represents the thirteenth prime number, that is also in equivalence with the sum of members in the aliquot tree (27, 13, 1, 0). ^{[3] [2]}
   • The next two larger numbers in the Collatz conjecture to require more than 111 steps to return to 1 are 54 and 55
   • Specifically, the fourteenth prime number 43 requires twenty-seven steps to reach 1.
   The sixth pair of twin primes is (41, 43), ^[6] whose respective prime indices generate a sum of 27.
2. Also,  36 = 6²  is the sum between PTNs  39 – 15 = 24  and  3 + 9 = 12. In this sequence, 111 is the seventh PTN.

References

1. "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved May 31, 2016.
2. Sloane, N. J. A. (ed.). "SequenceA001065(Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved October 31, 2023.
3. Sloane, N. J. A., ed. (January 11, 1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved October 31, 2023.
4. Sloane, N. J. A. (ed.). "SequenceA112695(Number of steps needed to reach 4,2,1 in Collatz' 3*n+1 conjecture.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved October 31, 2023.
5. Sloane, N. J. A. (ed.). "SequenceA286380(a(n) is the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R (A139391) is defined as R(k) equal to (3k+1)/2^r, with r as large as possible.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved November 8, 2023.
6. Sloane, N. J. A. (ed.). "SequenceA077800(List of twin primes {p, p+2}.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved November 8, 2023.
7. Sloane, N. J. A. (ed.). "SequenceA082897(Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved November 2, 2023.
8. 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.
9. Baez, John Carlos (February 15, 2016). "27 Lines on a Cubic Surface". AMS Blogs. American Mathematical Society . Retrieved October 31, 2023.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. 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.
19. "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
20. "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
21. Dave Andersen. "The Pi-Search Page". angio.net. Retrieved October 31, 2023.
22. Sloane, N. J. A. (ed.). "SequenceA064810(Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 1 is the 0th digit.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved October 31, 2023.
23. "Dark Energy, Dark Matter | Science Mission Directorate". science.nasa.gov. Retrieved November 8, 2020.
24. Steve Jenkins, Bones (2010), ISBN   978-0-545-04651-0
25. "Catalog of Solar Eclipses of Saros 27". NASA Eclipse Website. NASA. Retrieved February 27, 2022.
26. "Catalog of Lunar Eclipses in Saros 27". NASA Eclipse Website. NASA. Retrieved February 27, 2022.
27. "SpanishDict Grammar Guide". SpanishDict . Retrieved August 19, 2020.

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