56 (number)

← 55 56 57 →
← 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-six
Ordinal	56th (fifty-sixth)
Factorization	23 × 7
Divisors	1, 2, 4, 7, 8, 14, 28, 56
Greek numeral	ΝϚ´
Roman numeral	LVI, lvi
Binary	1110002
Ternary	20023
Senary	1326
Octal	708
Duodecimal	4812
Hexadecimal	3816

56 (fifty-six) is the natural number following 55 and preceding 57.

Mathematics

Regular 56-gon, associated by the Pythagoreans with Typhon

56 is:

• The sum of the first six triangular numbers (making it a tetrahedral number). ^[1]
• The number of ways to choose 3 out of 8 objects or 5 out of 8 objects, if order does not matter.
• The sum of six consecutive primes (3 + 5 + 7 + 11 + 13 + 17)
• a tetranacci number ^[2] and as a multiple of 7 and 8, a pronic number. ^[3] It is one of a few pronic numbers whose digits in decimal also are successive (5 and 6).
• a refactorable number, since 8 is one of its 8 divisors.
• The sum of the sums of the divisors of the first 8 positive integers. ^[4]
• A semiperfect number, since 56 is twice a perfect number.
• A partition number – the number of distinct ways 11 can be represented as the sum of natural numbers.
• An Erdős–Woods number, since it is possible to find sequences of 56 consecutive integers such that each inner member shares a factor with either the first or the last member. ^[5]
• The only known number n such that φ(n − 1)σ(n − 1) = φ(n)σ(n) = φ(n + 1)σ(n + 1), where φ(m) is Euler's totient function and σ(n) is the sum of the divisor function, see OEIS:  A244439 .
• The maximum determinant in an 8 by 8 matrix of zeroes and ones.
• The number of polygons formed by connecting all the 8 points on the perimeter of a two-times-two-square by straight lines. ^[6]

Plutarch ^[7] states that the Pythagoreans associated a polygon of 56 sides with Typhon and that they associated certain polygons of smaller numbers of sides with other figures in Greek mythology. While it is impossible to construct a perfect regular 56-sided polygon using a compass and straightedge, a close approximation has recently been discovered which it is claimed ^[8] might have been used at Stonehenge, and it is constructible if the use of an angle trisector is allowed since 56 = 2³ × 7. ^[9]

Organizations

• The symbol of the Hungarian Revolution of 1956.
• Brazilian politician Enéas Carneiro has an odd way of repeating the number of his party, "Fifty-Six" (cinquenta e seis, in Portuguese), making it a widely repeated jargon in his country.

Cosmogony

• According to Aristotle, 56 is the number of layers of the Universe – Earth plus 55 crystalline spheres above it. ^[10]

References

1. Sloane, N. J. A. (ed.). "SequenceA000292(Tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30.
2. Sloane, N. J. A. (ed.). "SequenceA000078(Tetranacci numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30.
3. Sloane, N. J. A. (ed.). "SequenceA002378(Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30.
4. Sloane, N. J. A. (ed.). "SequenceA024916(sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA059756(Erdős-Woods numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30.
6. Sloane, N. J. A. (ed.). "SequenceA255011". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-05-09.
7. Plutarch, Moralia V: 30
8. Pegs and Ropes: Geometry at Stonehenge
9. "Constructibility of Regular Polygons" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.
10. Heaven by Lisa Miller, (2010), ISBN   978-0-06-055475-0 - page 13.