Natural number
Cardinal eighty-four Ordinal 84th Factorization 22 × 3 × 7 Divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 (12) Greek numeral ΠΔ´ Roman numeral LXXXIV ,lxxxiv Binary 10101002 Ternary 100103 Senary 2206 Octal 1248 Duodecimal 7012 Hexadecimal 5416
84 (eighty-four ) is thenatural number following83 and preceding85 . It is sevendozens .
Ahepteract is a seven-dimensional hypercube with 84penteract 5-faces. 84 is asemiperfect number ,[ 1] twin primes ( 41 + 43 ) {\displaystyle (41+43)} [ 2] perfect powers indecimal .[ 3]
It is the third (or the 2)dodecahedral number ,[ 4] triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventhtetrahedral number .[ 5]
The number ofdivisors of 84 is 12.[ 6] largely composite number .[ 7]
The twenty-secondunique prime indecimal , with notably differentdigits than its preceding (and known following) terms in the samesequence , contains a total of 84 digits.[ 8]
Ahepteract is a seven-dimensional hypercube with 84penteract 5-faces.[ 9]
84 is thelimit superior of the largest finite subgroup of themapping class group of agenus g {\displaystyle g} g {\displaystyle g} [citation needed
UnderHurwitz's automorphisms theorem , a smooth connectedRiemann surface X {\displaystyle X} genus g > 1 {\displaystyle g>1} automorphism group A u t ( X ) = G {\displaystyle \mathrm {Aut} (X)=G} order is classically bound to| G | ≤ 84 ( g − 1 ) {\displaystyle |G|\leq 84{\text{ }}(g-1)} [ 10]
84 is the thirtieth and largestn {\displaystyle n} cyclotomic field Q ( ζ n ) {\displaystyle \mathrm {Q} (\zeta _{n})} 1 {\displaystyle 1} factorization ), preceding60 (that is thecomposite index of 84),[ 11] 48 .[ 12] [ 13]
There are 84zero divisors in the 16-dimensionalsedenions S {\displaystyle \mathbb {S} } [ 14]
84 is also:
^ Sloane, N. J. A. (ed.)."Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ Sloane, N. J. A. (ed.)."Sequence A077800 (List of twin primes {p, p+2})" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-06-08 .^ Sloane, N. J. A. (ed.)."Sequence A075308 (Number of n-digit perfect powers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ Sloane, N. J. A. (ed.)."Sequence A006566 (Dodecahedral numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ Sloane, N. J. A. (ed.)."Sequence A000292 (Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ Sloane, N. J. A. (ed.)."Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ Sloane, N. J. A. (ed.)."Sequence A067128 (Ramanujan's largely composite numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ 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))" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-06-08 .^ Sloane, N. J. A. (ed.)."Sequence A046092 (4 times triangular numbers: a(n) = 2*n*(n+1))" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ 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)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ 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))" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ 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). RetrievedSeptember 29, 2022 .
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