84 (number)

Template:Infobox number 84 (eighty-four) is the natural number following 83 and preceding 85. It is seven dozens.

In mathematics

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

84 is a semiperfect number, being thrice a perfect number, and the sum of the sixth pair of twin primes $(41+43)$. It is the number of four-digit perfect powers in decimal.

It is the third (or second) dodecahedral number, and the sum of the first seven triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventh tetrahedral number.

The number of divisors of 84 is 12. As no smaller number has more than 12 divisors, 84 is a largely composite number.

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.

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 $g$ surface divided by $g$.Template:Cn

Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface $X$ of genus $g>1$ will contain an automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(X)=G$ whose order is classically bound to $|G|\le 84(g-1)$.

84 is the thirtieth and largest $n$ for which the cyclotomic field $\mathrm{Q}({\zeta }_n)$ has class number $1$ (or unique factorization), preceding 60 (that is the composite index of 84), and 48.

There are 84 zero divisors in the 16-dimensional sedenions $𝕊$.

In other fields

Eighty-four is also:

• The number of years in the Template:Lang, a cycle used in the past by Celtic peoples, equal to 3 cycles of the Julian Calendar and to 4 Metonic cycles and 1 octaeterisTemplate:Relevance inline