58 (number)

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(Redirected from Fifty-eight)
← 57 58 59 →
Cardinalfifty-eight
Ordinal58th
(fifty-eighth)
Factorization2 × 29
Divisors1, 2, 29, 58
Greek numeralΝΗ´
Roman numeralLVIII
Binary1110102
Ternary20113
Senary1346
Octal728
Duodecimal4A12
Hexadecimal3A16

58 (fifty-eight) is the natural number following 57 and preceding 59.

Mathematics[edit]

Fifty-eight is the seventeenth discrete semiprime[1] and the ninth with 2 as the lowest non-unitary divisor; thus of the form , where is a higher prime.

58 is equal to the sum of the first seven consecutive prime numbers:[2]

This is a difference of 1 from the seventeenth prime number and seventh super-prime, 59.[3][4] Furthermore, is semiprime (the second such number for after 2).[5]

Fifty-eight is an 11-gonal number, after 30 (and 11),[6] and it is a Smith number.[7] Also:

There is no solution to the equation , making fifty-eight a noncototient.[13] However, the totient summatory function over the first thirteen integers is 58.[14]

The regular icosahedron produces fifty-eight distinct stellations, the most of any other Platonic solid, which collectively produce sixty-two stellations.[15][16]

Coxeter groups[edit]

With regard to Coxeter groups and uniform polytopes in higher dimensional spaces, there are:

  • 58 distinct uniform polytopes in the fifth dimension that are generated from symmetries of three Coxeter groups, they are the A5 simplex group, B5 cubic group, and the D5 demihypercubic group;
  • 58 fundamental Coxeter groups that generate uniform polytopes in the seventh dimension, with only four of these generating uniform non-prismatic figures.

There exist 58 total paracompact Coxeter groups of ranks four through ten, with realizations in dimensions three through nine. These solutions all contain infinite facets and vertex figures, in contrast from compact hyperbolic groups that contain finite elements; there are no other such groups with higher or lower ranks.

In mythology[edit]

Belief in the existence of 58 original sins by several civilizations native to Central America or South America caused the number to symbolize misfortune. Aztec oracles supposedly stumbled across the number an unnaturally high number of times before disaster fell. One famous recording of this, though largely discredited as mere folktale, concerned the oracle of Moctezuma II, who allegedly counted 58 pieces of gold scattered before a sacrificial pit the day prior to the arrival of Hernán Cortés.[citation needed]

Other fields[edit]

Base Hexxagōn starting grid, with fifty-eight "usable" cells

58 is the number of usable cells on a Hexxagon game board.

References[edit]

  1. ^ Sloane, N. J. A. (ed.). "Sequence A001358". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A007504 (Sum of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-20.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-20.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-20.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A104494 (Positive integers n such that n^17 + 1 is semiprime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
  6. ^ "Sloane's A051682 : 11-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  7. ^ "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
  10. ^ Sloane, N. J. A., ed. (1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved 2024-02-27.
  11. ^ "Sloane's A013646: Least m such that continued fraction for sqrt(m) has period n". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-03-18.
  12. ^ "Sloane's A028442 : Numbers n such that Mertens' function is zero". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  13. ^ "Sloane's A005278 : Noncototients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  14. ^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
  15. ^ H. S. M. Coxeter; P. Du Val; H. T. Flather; J. F. Petrie (1982). The Fifty-Nine Icosahedra. New York: Springer. doi:10.1007/978-1-4613-8216-4. ISBN 978-1-4613-8216-4.
  16. ^ Webb, Robert. "Enumeration of Stellations". Stella. Archived from the original on 2022-11-26. Retrieved 2023-01-18.