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84 (number)

Natural number From Wikipedia, the free encyclopedia

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

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In mathematics

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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 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 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 .[10]

84 is the thirtieth and largest for which the cyclotomic field has class number (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 .[14]

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In other fields

Eighty-four is also:

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