84 (number)

← 83 84 85 →
← 80 81 82 83 84 85 86 87 88 89 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	eighty-four
Ordinal	84th (eighty-fourth)
Factorization	2^2 × 3 × 7
Divisors	1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Greek numeral	ΠΔ´
Roman numeral	LXXXIV
Binary	10101002
Ternary	100103
Senary	2206
Octal	1248
Duodecimal	7012
Hexadecimal	5416

84 (eighty-four) is the natural number following 83 and preceding 85. It is seven dozens.

In mathematics

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.

84 is a semiperfect number,^[1] being thrice a perfect number, and the sum of the sixth pair of twin primes ${\displaystyle (41+43)}$.^[2] It is the number of four-digit perfect powers in decimal.^[3]

It is the third (or second) dodecahedral number,^[4] and the sum of the first seven triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventh tetrahedral number.^[5]

The twenty-second unique prime in decimal, with notably different digits than its preceding (and known following) terms in the same sequence, contains a total of 84 digits.^[6]

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.^[7]

84 is the limit superior of the largest finite subgroup of the mapping class group of a genus ${\displaystyle g}$ surface divided by ${\displaystyle g}$.^{[citation needed]}

Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface ${\displaystyle X}$ of genus ${\displaystyle g>1}$ will contain an automorphism group ${\displaystyle \mathrm {Aut} (X)=G}$ whose order is classically bound to ${\displaystyle |G|\leq 84{\text{ }}(g-1)}$.^[8]

84 is the thirtieth and largest ${\displaystyle n}$ for which the cyclotomic field ${\displaystyle \mathrm {Q} (\zeta _{n})}$ has class number ${\displaystyle 1}$ (or unique factorization), preceding 60 (that is the composite index of 84),^[9] and 48.^[10][11]

There are 84 zero divisors in the 16-dimensional sedenions ${\displaystyle \mathbb {S} }$.^[12]

In astronomy

• Messier object M84, a magnitude 11.0 lenticular galaxy in the constellation Virgo
• The New General Catalogue object NGC 84, a single star in the constellation Andromeda

In other fields

dial +84 for Vietnam

Eighty-four is also:

• The year AD 84, 84 BC, or 1984.
• The number of years in the Insular latercus, a cycle used in the past by Celtic peoples,^[13] equal to 3 cycles of the Julian Calendar and to 4 Metonic cycles and 1 octaeteris
• The atomic number of polonium
• The model number of Harpoon missile
• WGS 84 - The latest revision of the World Geodetic System, a fixed global reference frame for the Earth.
• The house number of 84 Avenue Foch
• The number of the French department Vaucluse
• The code for international direct dial phone calls to Vietnam
• The town of Eighty Four, Pennsylvania
• The company 84 Lumber
• The ISBN Group Identifier for books published in Spain
• A variation of the game 42 played with two sets of dominoes.
• The film 84 Charing Cross Road (1987) starring Anne Bancroft and Anthony Hopkins
• KKNX Radio 84 in Eugene, Oregon
• The B-Side to "Up All Night" (Take That song)
• British Army term for the 84mm Carl Gustav recoilless rifle.
• How many Earth years it takes Uranus to orbit the Sun once

See also

• List of highways numbered 84

References

1. Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2})". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-08.
3. Sloane, N. J. A. (ed.). "Sequence A075308 (Number of n-digit perfect powers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. Sloane, N. J. A. (ed.). "Sequence A006566 (Dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. Sloane, N. J. A. (ed.). "Sequence A000292 (Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Sloane, N. J. A. (ed.). "Sequence A040017 (Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-08.
7. Sloane, N. J. A. (ed.). "Sequence A046092 (4 times triangular numbers: a(n) = 2*n*(n+1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. Giulietti, Massimo; Korchmaros, Gabor (2019). "Algebraic curves with many automorphisms". Advances in Mathematics. 349 (9). Amsterdam, NL: Elsevier: 162–211. arXiv:1702.08812. doi:10.1016/J.AIM.2019.04.003. MR 3938850. S2CID 119269948. Zbl 1419.14040.
9. Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. Vol. 83 (2nd ed.). Springer-Verlag. pp. 205–206 (Theorem 11.1). ISBN 0-387-94762-0. MR 1421575. OCLC 34514301. Zbl 0966.11047.
11. Sloane, N. J. A. (ed.). "Sequence A005848 (Cyclotomic fields with class number 1 (or with unique factorization))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
12. Cawagas, Raoul E. (2004). "On the Structure and Zero Divisors of the Cayley-Dickson Sedenion Algebra". Discussiones Mathematicae – General Algebra and Applications. 24 (2). PL: University of Zielona Góra: 262–264. doi:10.7151/DMGAA.1088. MR 2151717. S2CID 14752211. Zbl 1102.17001.
13. Venerabilis, Beda (May 13, 2020) [731 AD]. "Historia Ecclesiastica gentis Anglorum/Liber Secundus" [The Ecclesiastical History of the English Nation/Second Book]. Wikisource (in Latin). Retrieved September 29, 2022.