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	22 × 3 × 7
Divisors	1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Greek numeral	ΠΔ´
Roman numeral	LXXXIV, 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 number of divisors of 84 is 12. ^[6] As no smaller number has more than 12 divisors, 84 is a largely composite number. ^[7]

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. ^[8]

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces. ^[9]

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 ${\displaystyle |G|\leq 84{\text{ }}(g-1)}$. ^[10]

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), ^[11] and 48. ^{[12] [13]}

There are 84 zero divisors in the 16-dimensional sedenions ${\displaystyle \mathbb {S} }$. ^[14]

In other fields

Eighty-four is also:

• The number of years in the Insular latercus , a cycle used in the past by Celtic peoples, ^[15] equal to 3 cycles of the Julian Calendar and to 4 Metonic cycles and 1 octaeteris ^{[ relevant? ]}

References

1. Sloane, N. J. A. (ed.). "SequenceA005835(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.). "SequenceA077800(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.). "SequenceA075308(Number of n-digit perfect powers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
4. Sloane, N. J. A. (ed.). "SequenceA006566(Dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA000292(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.). "SequenceA000005(d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
7. Sloane, N. J. A. (ed.). "SequenceA067128(Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Sloane, N. J. A. (ed.). "SequenceA040017(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.
9. Sloane, N. J. A. (ed.). "SequenceA046092(4 times triangular numbers: a(n) = 2*n*(n+1))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
10. 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.
11. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
12. 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.
13. Sloane, N. J. A. (ed.). "SequenceA005848(Cyclotomic fields with class number 1 (or with unique factorization))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
14. 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.
15. 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.