58 (number)

← 57 58 59 →
← 50 51 52 53 54 55 56 57 58 59 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	fifty-eight
Ordinal	58th (fifty-eighth)
Factorization	2 × 29
Divisors	1, 2, 29, 58
Greek numeral	ΝΗ´
Roman numeral	LVIII
Binary	1110102
Ternary	20113
Senary	1346
Octal	728
Duodecimal	4A12
Hexadecimal	3A16

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

Mathematics

Fifty-eight is the seventeenth discrete semiprime ^[1] and the ninth with 2 as the lowest non-unitary divisor; thus of the form ${\displaystyle 2\times q}$, where ${\displaystyle q}$ is a higher prime.

58 is equal to the sum of the first seven consecutive prime numbers: ^[2] ${\displaystyle 2+3+5+7+11+13+17=58.}$

This is a difference of 1 from the seventeenth prime number and seventh super-prime, 59. ^{[3] [4]} Furthermore, ${\displaystyle 58^{17}+1}$ is semiprime (the second such number ${\displaystyle n}$ for ${\displaystyle n^{17}+1,}$ after 2). ^[5]

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

• 58 is the second member of the fifth cluster of two semiprimes (57, 58), following (25, 26) and preceding (118, 119). ^[8]
• 58 has an aliquot sum of 32 ^[9] within an aliquot sequence of two composite numbers (58, 32, 13, 1, 0) in the 13-aliquot tree. ^[10]
• 58 is the smallest integer in decimal whose square root has a continued fraction with period 7. ^[11]
• Given 58, the Mertens function returns ${\displaystyle 0}$. ^[12]

There is no solution to the equation ${\displaystyle x-\varphi (x)=58}$, 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

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

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

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

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

References

1. Sloane, N. J. A. (ed.). "SequenceA001358". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA007504(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.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-20.
4. Sloane, N. J. A. (ed.). "SequenceA006450(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.). "SequenceA104494(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.). "SequenceA001358(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.). "SequenceA001065(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.). "SequenceA002088(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.