84 (number)

84 is a semiperfect number,^[1] being thrice a perfect number, and the sum of the sixth pair of twin primes ${\displaystyle (41+43)}$.^[2] It is the number of four-digit perfect powers in decimal.^[3]

It is the third (or the 2) dodecahedral number,^[4] and the sum of the first seven triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventh tetrahedral number.^[5]

The number of divisors of 84 is 12.^[6] As no smaller number has more than 12 divisors, 84 is a largely composite number.^[7]

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.^[8]

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.^[9]

84 is the limit superior of the largest finite subgroup of the mapping class group of a genus ${\displaystyle g}$ surface divided by ${\displaystyle g}$.^{[citation needed]}

Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface ${\displaystyle X}$ of genus ${\displaystyle g>1}$ will contain an automorphism group ${\displaystyle \mathrm {Aut} (X)=G}$ whose order is classically bound to ${\displaystyle |G|\leq 84{\text{ }}(g-1)}$.^[10]

84 is the thirtieth and largest ${\displaystyle n}$ for which the cyclotomic field ${\displaystyle \mathrm {Q} (\zeta _{n})}$ has class number ${\displaystyle 1}$ (or unique factorization), preceding 60 (that is the composite index of 84),^[11] and 48.^[12][13]

There are 84 zero divisors in the 16-dimensional sedenions ${\displaystyle \mathbb {S} }$.^[14]

84 is also:

• The number of years in the Insular latercus, a cycle used in the past by Celtic peoples,^[15] equal to 3 cycles of the Julian Calendar and to 4 Metonic cycles and 1 octaeteris^[relevant?]
• The international calling code for Vietnam