32 (number)

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31 32 33
Cardinal thirty-two
Ordinal 32nd
(thirty-second)
Factorization 25
Divisors 1, 2, 4, 8, 16, 32
Greek numeral ΛΒ´
Roman numeral XXXII
Binary 1000002
Ternary 10123
Senary 526
Octal 408
Duodecimal 2812
Hexadecimal 2016

32 (thirty-two) is the natural number following 31 and preceding 33.

Contents

Mathematics

32 is the fifth power of two (), making it the first non-unitary fifth-power of the form where is prime. 32 is the totient summatory function over the first 10 integers, [1] and the smallest number with exactly 7 solutions for .

The aliquot sum of a power of two is always one less than the number itself, therefore the aliquot sum of 32 is 31. [2]

The product between neighbor numbers of 23 , the dual permutation of the digits of 32 in decimal, is equal to the sum of the first 32 integers: . [3] [a]

32 is also a Leyland number expressible in the form , where: [5] [b]

The eleventh Mersenne number is the first to have a prime exponent (11) that does not yield a Mersenne prime, equal to: [7] [c]

When read in binary, the first 32 rows of Pascal's Triangle represent the thirty-two divisors that belong to the largest constructible polygon. Binary Pascal's Triangle.png
When read in binary, the first 32 rows of Pascal's Triangle represent the thirty-two divisors that belong to the largest constructible polygon.

The product of the five known Fermat primes is equal to the number of sides of the largest regular constructible polygon with a straightedge and compass that has an odd number of sides, with a total of sides numbering

The first 32 rows of Pascal's triangle read as single binary numbers represent the 32 divisors that belong to this number, which is also the number of sides of all odd-sided constructible polygons with simple tools alone (if the monogon is also included). [10]

There are also a total of 32 uniform colorings to the 11 regular and semiregular tilings. [11]

There are 32 three-dimensional crystallographic point groups [12] and 32 five-dimensional crystal families, [13] and the maximum determinant in a 7 by 7 matrix of only zeroes and ones is 32. [14] In sixteen dimensions, the sedenions generate a non-commutative loop of order 32, [15] and in thirty-two dimensions, there are at least 1,160,000,000 even unimodular lattices (of determinants 1 or −1); [16] which is a marked increase from the twenty-four such Niemeier lattices that exists in twenty-four dimensions, or the single lattice in eight dimensions (these lattices only exist for dimensions ). Furthermore, the 32nd dimension is the first dimension that holds non-critical even unimodular lattices that do not interact with a Gaussian potential function of the form of root and . [17]

32 is the furthest point in the set of natural numbers where the ratio of primes (2, 3, 5, ..., 31) to non-primes (0, 1, 4, ..., 32) is [d]

The trigintaduonions form a 32-dimensional hypercomplex number system. [20]

In science

Astronomy

In music

In religion

In the Kabbalah, there are 32 Kabbalistic Paths of Wisdom. This is, in turn, derived from the 32 times of the Hebrew names for God, Elohim appears in the first chapter of Genesis.

One of the central texts of the Pāli Canon in the Theravada Buddhist tradition, the Digha Nikaya, describes the appearance of the historical Buddha with a list of 32 physical characteristics.

The Hindu scripture Mudgala Purana also describes Ganesha as taking 32 forms.

In sports

In other fields

Thirty-two could also refer to:

Notes

  1. 32 is the ninth 10-happy number, while 23 is the sixth. [4] Their sum is 55, which is the tenth triangular number, [3] while their difference is
  2. On the other hand, a regular 32-sided triacontadigon contains distinct symmetries. [6]
    For comparison, a 16-sided hexadecagon contains 14 symmetries, an 8-sided octagon contains 11 symmetries, and a square contains 8 symmetries.
  3. Specifically, 31 is the eleventh prime number, equal to the sum of 20 and its composite index 11, where 33 is the twenty-first composite number, equal to the sum of 21 and its composite index 12 (which are palindromic numbers). [8] [9] 32 is the only number to lie between two adjacent numbers whose values can be directly evaluated from sums of associated prime and composite indices (32 is the twentieth composite number, which maps to 31 through its prime index of 11, and 33 by a factor of 11, that is the composite index of 20; the aliquot part of 32 is 31 as well). [2] This is due to the fact that the ratio of composites to primes increases very rapidly, by the prime number theorem.
  4. 29 is the only earlier point, where there are twenty non primes, and ten primes. 40 — twice the composite index of 32 — lies between the 8th pair of sexy primes (37, 43), [18] which represent the only two points in the set of natural numbers where the ratio of prime numbers to composite numbers (up to) is 1/2. Where 68 is the forty-eighth composite, 48 is the thirty second, with the difference 6848 = 20 , the composite index of 32. [8] Otherwise, thirty-two lies midway between primes (23, 41), (17, 47) and (3, 61).
    At 33, there are 11 numbers that are prime and 22 that are not, when considering instead the set of natural numbers that does not include 0. The product 11 × 33 = 363 represents the thirty-second number to return 0 for the Mertens function M(n). [19]

Related Research Articles

15 (fifteen) is the natural number following 14 and preceding 16.

21 (twenty-one) is the natural number following 20 and preceding 22.

33 (thirty-three) is the natural number following 32 and preceding 34.

78 (seventy-eight) is the natural number following 77 and preceding 79.

70 (seventy) is the natural number following 69 and preceding 71.

90 (ninety) is the natural number following 89 and preceding 91.

23 (twenty-three) is the natural number following 22 and preceding 24.

25 (twenty-five) is the natural number following 24 and preceding 26.

35 (thirty-five) is the natural number following 34 and preceding 36

34 (thirty-four) is the natural number following 33 and preceding 35.

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

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

63 (sixty-three) is the natural number following 62 and preceding 64.

64 (sixty-four) is the natural number following 63 and preceding 65.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

168 is the natural number following 167 and preceding 169.

230 is the natural number following 229 and preceding 231.

240 is the natural number following 239 and preceding 241.

888 is the natural number following 887 and preceding 889.

14 (fourteen) is the natural number following 13 and preceding 15.

References

  1. Sloane, N. J. A. (ed.). "SequenceA002088(Sum of totient function)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-05-04.
  2. 1 2 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 2024-01-10.
  3. 1 2 Sloane, N. J. A. (ed.). "SequenceA000217(Triangular numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-05-04.
  4. "Sloane's A007770 : Happy numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  5. "Sloane's A076980 : Leyland numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  6. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "Chapter 20: Generalized Schaefli symbols (Types of symmetry of a polygon)". The Symmetries of Things (1st ed.). New York: CRC Press (Taylor & Francis). pp. 275–277. doi:10.1201/b21368. ISBN   978-1-56881-220-5. OCLC   181862605. Zbl   1173.00001.
  7. Sloane, N. J. A. (ed.). "SequenceA000225(a(n) equal to 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-08.
  8. 1 2 Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-08.
  9. Sloane, N. J. A. (ed.). "SequenceA00040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-08.
  10. Conway, John H.; Guy, Richard K. (1996). "The Primacy of Primes". The Book of Numbers. New York, NY: Copernicus (Springer). pp. 137–142. doi:10.1007/978-1-4612-4072-3. ISBN   978-1-4612-8488-8. OCLC   32854557. S2CID   115239655.
  11. Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.9 Archimedean and uniform colorings". Tilings and Patterns . New York: W. H. Freeman and Company. pp. 102–107. doi:10.2307/2323457. ISBN   0-7167-1193-1. JSTOR   2323457. OCLC   13092426. S2CID   119730123.
  12. Sloane, N. J. A. (ed.). "SequenceA004028(Number of geometric n-dimensional crystal classes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-08.
  13. Sloane, N. J. A. (ed.). "SequenceA004032(Number of n-dimensional crystal families.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-08.
  14. Sloane, N. J. A. (ed.). "SequenceA003432(Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-04-04.
  15. Cawagas, Raoul E.; Gutierrez, Sheree Ann G. (2005). "The Subloop Structure of the Cayley-Dickson Sedenion Loop" (PDF). Matimyás Matematika. 28 (1–3). Diliman, Q.C.: The Mathematical Society of the Philippines: 13–15. ISSN   0115-6926. Zbl   1155.20315.
  16. Baez, John C. (November 15, 2014). "Integral Octonions (Part 8)". John Baez's Stuff. U.C. Riverside, Department of Mathematics. Retrieved 2023-05-04.
  17. Heimendahl, Arne; Marafioti, Aurelio; et al. (June 2022). "Critical Even Unimodular Lattices in the Gaussian Core Model". International Mathematics Research Notices . 1 (6). Oxford: Oxford University Press: 5352. arXiv: 2105.07868 . doi:10.1093/imrn/rnac164. S2CID   234742712. Zbl   1159.11020.
  18. Sloane, N. J. A. (ed.). "SequenceA156274(List of prime pairs of the form (p, p+6).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-11.
  19. Sloane, N. J. A. (ed.). "SequenceA028442(Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-11.
  20. Saniga, Metod; Holweck, Frédéric; Pracna, Petr (2015). "From Cayley-Dickson Algebras to Combinatorial Grassmannians". Mathematics. 3 (4). MDPI AG: 1192–1221. arXiv: 1405.6888 . doi: 10.3390/math3041192 . ISSN   2227-7390.