← 83 84 85 →
Cardinaleighty-four
Ordinal84th
(eighty-fourth)
Factorization22 × 3 × 7
Divisors1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Greek numeralΠΔ´
Roman numeralLXXXIV
Binary10101002
Ternary100103
Senary2206
Octal1248
Duodecimal7012

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

In mathematics

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 other fields

Eighty-four is also:

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. ^
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. ^
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.