From Wikipedia, the free encyclopedia
Natural number
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
Binary 10101002
Ternary 100103
Senary 2206
Octal 1248
Duodecimal 7012
Hexadecimal 5416
84 (eighty-four ) is the
natural number following
83 and preceding
85 .
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
(
41
+
43
)
{\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, 36), which makes it the sixth
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.
84 is the
limit superior of the largest finite subgroup of the
mapping class group of a
genus
g
{\displaystyle g}
surface divided by
g
{\displaystyle g}
.[
citation needed ]
Under
Hurwitz's automorphisms theorem , a smooth connected
Riemann surface
X
{\displaystyle X}
of
genus
g
>
1
{\displaystyle g>1}
will contain an
automorphism group
A
u
t
(
X
)
=
G
{\displaystyle \mathrm {Aut} (X)=G}
whose
order is classically bound to
|
G
|
≤
84
(
g
−
1
)
{\displaystyle |G|\leq 84{\text{ }}(g-1)}
.
[7]
84 is the thirtieth and largest
n
{\displaystyle n}
for which the
cyclotomic field
Q
(
ζ
n
)
{\displaystyle \mathrm {Q} (\zeta _{n})}
has class number
1
{\displaystyle 1}
(or unique
factorization ), preceding
60 (that is the
composite index of 84),
[8] and
48 .
[9]
[10]
There are 84
zero divisors in the 16-dimensional
sedenions
S
{\displaystyle \mathbb {S} }
.
[11]
In astronomy
In other fields
dial +84 for
Vietnam
Eighty-four is also:
See also
References
^
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. Retrieved 2023-06-08 .
^
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 .
^
Sloane, N. J. A. (ed.).
"Sequence A075308 (Number of n-digit perfect powers.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-10 .
^
Sloane, N. J. A. (ed.).
"Sequence A006566 (Dodecahedral numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-08 .
^
"Sloane's A000292 : Tetrahedral numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved May 29, 2016 .
^
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 .
^ 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 .
^
Sloane, N. J. A. (ed.).
"Sequence A002808 (The composite numbers.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-03-20 .
^
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 .
^
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. Retrieved 2024-03-20 .
^ 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 .
^ 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 .
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