84 (number)

In mathematics

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 the 2) 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.

Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface X of genus g 1 will contain an automorphism group \mathrm{Aut}(X) = G whose order is classically bound to |G| \le 84 \text { } (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 \mathbb S.

In other fields

84 is also:

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

References

References

1. {{Cite OEIS. A005835. Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n
2. {{Cite OEIS. A077800. List of twin primes {p, p+2}
3. {{Cite OEIS. A075308. Number of n-digit perfect powers
4. {{Cite OEIS. A006566. Dodecahedral numbers
5. {{Cite OEIS. A000292
6. {{Cite OEIS. A000005. d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
7. {{Cite OEIS. A067128. Ramanujan's largely composite numbers
8. {{Cite OEIS. A040017. Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime)
9. {{Cite OEIS. A046092
10. (2019). "Algebraic curves with many automorphisms". [[Elsevier]].
11. {{Cite OEIS. A002808. The composite numbers
12. Washington, Lawrence C.. (1997). "Introduction to Cyclotomic Fields". [[Springer-Verlag]].
13. {{Cite OEIS. A005848. Cyclotomic fields with class number 1 (or with unique factorization)
14. Cawagas, Raoul E.. (2004). "On the Structure and Zero Divisors of the Cayley-Dickson Sedenion Algebra". [[University of Zielona Góra]].
15. Venerabilis, Beda. (May 13, 2020). "Historia Ecclesiastica gentis Anglorum/Liber Secundus".