84 (number)

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← 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
Hexadecimal5416

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

In mathematics[edit]

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 .[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 surface divided by .[citation needed]

Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface of genus will contain an automorphism group whose order is classically bound to .[7]

There are 84 zero divisors in the 16-dimensional sedenions .[8]

In astronomy[edit]

In other fields[edit]

dial +84 for Vietnam

Eighty-four is also:

See also[edit]

References[edit]

  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. Retrieved 2023-06-08.
  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. Retrieved 2024-02-10.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A006566 (Dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-08.
  5. ^ "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 29, 2016.
  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. ^ Giulietti, Massimo; Korchmaros, Gabor (2019). "Algebraic curves with many automorphisms". Advances in Mathematics. Amsterdam, NL: Elsevier. 349 (9): 162–211. arXiv:1702.08812. doi:10.1016/J.AIM.2019.04.003. MR 3938850. S2CID 119269948. Zbl 1419.14040.
  8. ^ Cawagas, Raoul E. (2004). "On the Structure and Zero Divisors of the Cayley-Dickson Sedenion Algebra". Discussiones Mathematicae – General Algebra and Applications. PL: University of Zielona Góra. 24 (2): 262–264. doi:10.7151/DMGAA.1088. MR 2151717. S2CID 14752211. Zbl 1102.17001.
  9. ^ 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.