← 31 32 33 →
Cardinalthirty-two
Ordinal32nd
(thirty-second)
Factorization25
Divisors1, 2, 4, 8, 16, 32
Greek numeralΛΒ´
Roman numeralXXXII
Binary1000002
Ternary10123
Senary526
Octal408
Duodecimal2812

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

## Mathematics

32 is the fifth power of two (${\displaystyle 2^{5))$), making it the first non-unitary fifth-power of the form ${\displaystyle p^{5))$ where ${\displaystyle p}$ is prime. 32 is the totient summatory function ${\displaystyle \Phi (n)}$ over the first 10 integers,[1] and the smallest number ${\displaystyle n}$ with exactly 7 solutions for ${\displaystyle \varphi (n)}$.

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]

{\displaystyle {\begin{aligned}32&=1^{1}+2^{2}+3^{3}\\32&=(1\times 4)+(2\times 5)+(3\times 6)\\32&=(1\times 2)+(1\times 2\times 3)+(1\times 2\times 3\times 4)\\\end{aligned))}

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: ${\displaystyle 22\times 24=528}$.[3][a]

32 is also a Leyland number expressible in the form ${\displaystyle x^{y}+y^{x))$, where:[5][b]

${\displaystyle 32=2^{4}+4^{2}.}$

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

${\displaystyle 2047=32^{2}+(31\times 33)=1024+1023=2^{11}-1.}$

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

${\displaystyle 2^{32}-1=3\cdot 5\cdot 17\cdot 257\cdot 65\;537=4\;294\;967\;295.}$

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 ${\displaystyle \mathbb {S} _{L))$ 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 ${\displaystyle \mathrm {E} _{8))$ lattice in eight dimensions (these lattices only exist for dimensions ${\displaystyle d\propto 8}$). 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 ${\displaystyle f_{\alpha }(r)=e^{-\alpha {r))}$ of root ${\displaystyle r}$ and ${\displaystyle \alpha >0}$.[17]

32 is the furthest point in the set of natural numbers ${\displaystyle \mathbb {N} _{0))$ where the ratio of primes (2, 3, 5, ..., 31) to non-primes (0, 1, 4, ... 32) is ${\displaystyle {\tfrac {1}{2)).}$[d]

## 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 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 ${\displaystyle 9=3^{2}.}$
2. ^ On the other hand, a regular 32-sided icosidodecagon contains ${\displaystyle 2^{3}+3^{2}=17}$ 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. Note, that 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 ${\displaystyle \mathbb {N} ^{+))$ 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]

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-04.
2. ^ a b 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 2024-01-10.
3. ^ a b Sloane, N. J. A. (ed.). "Sequence A000217 (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.). "Sequence A000225 (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. ^ a b Sloane, N. J. A. (ed.). "Sequence A002808 (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.). "Sequence A00040 (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.). "Sequence A004028 (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.). "Sequence A004032 (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.). "Sequence A003432 (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. Diliman, Q.C.: The Mathematical Society of the Philippines. 28 (1–3): 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. Oxford: Oxford University Press. 1 (6): 5352. arXiv:2105.07868. doi:10.1093/imrn/rnac164. S2CID 234742712. Zbl 1159.11020.
18. ^ Sloane, N. J. A. (ed.). "Sequence A156274 (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.). "Sequence A028442 (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.