84 (number)

Template:Infobox number 84 (eighty-four) is the natural number following 82 and preceding 86. It is seven dozens.

In mathematics

File:Hepteract ortho petrie.svg

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.

84 is a semiperfect number,<ref>Template:Cite OEIS</ref> being thrice a perfect number, and the sum of the sixth pair of twin primes <math>(41 + 43)</math>.<ref>Template:Cite OEIS</ref> It is the number of four-digit perfect powers in decimal.<ref>Template:Cite OEIS</ref>

It is the third (or the 2) dodecahedral number,<ref>Template:Cite OEIS</ref> and the sum of the first seven triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventh tetrahedral number.<ref>Template:Cite OEIS</ref>

The number of divisors of 84 is 12.<ref name="OEIS-A000005">Template:Cite OEIS</ref> As no smaller number has more than 12 divisors, 84 is a largely composite number.<ref name="OEIS-A067128">Template:Cite OEIS</ref>

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.<ref>Template:Cite OEIS</ref>

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.<ref>Template:Cite OEIS</ref>

84 is the limit superior of the largest finite subgroup of the mapping class group of a genus <math>g</math> surface divided by <math>g</math>.Template:Cn

Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface <math>X</math> of genus <math>g > 1</math> will contain an automorphism group <math>\mathrm{Aut}(X) = G</math> whose order is classically bound to <math>|G| \le 84 \text { } (g - 1)</math>.<ref>Template:Cite journal</ref>

84 is the thirtieth and largest <math>n</math> for which the cyclotomic field <math>\mathrm {Q}(\zeta_{n})</math> has class number <math>1</math> (or unique factorization), preceding 60 (that is the composite index of 84),<ref>Template:Cite OEIS</ref> and 48.<ref>Template:Cite book</ref><ref>Template:Cite OEIS</ref>

There are 84 zero divisors in the 16-dimensional sedenions <math>\mathbb S</math>.<ref>Template:Cite journal</ref>

In other fields

Eighty-four is also:

• The number of years in the {{#invoke:Lang|lang}}, a cycle used in the past by Celtic peoples,<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> equal to 3 cycles of the Julian Calendar and to 4 Metonic cycles and 1 octaeterisTemplate:Relevance inline