32 (number)

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

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

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

$$ \begin{align} 32 & = 1^{1} + 2^{2} + 3^{3} \\ 32 & = (1\times4)+(2\times5)+(3\times6) \\ 32 & = (1\times2)+(1\times2\times3)+(1\times2\times3\times4) \\ \end{align}$$

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: $$22 \times 24= 528$$.

32 is also a Leyland number expressible in the form $$x^y + y^x$$, where: $$32=2^{4} + 4^{2}.$$

The eleventh Mersenne number is the first to have a prime exponent (11) that does not yield a Mersenne prime, equal to: $$2047 = 32^{2} + (31 \times 33) = 1024 + 1023 = 2^{11} - 1.$$



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 $$2^{32} - 1 = 3\cdot5\cdot17\cdot257\cdot65\;537 = 4\;294\;967\;295.$$

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).

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

There are 32 three-dimensional crystallographic point groups and 32 five-dimensional crystal families, and the maximum determinant in a 7 by 7 matrix of only zeroes and ones is 32. In sixteen dimensions, the sedenions generate a non-commutative loop $$\mathbb {S}_{L}$$ of order 32, and in thirty-two dimensions, there are at least 1,160,000,000 even unimodular lattices (of determinants 1 or −1); which is a marked increase from the twenty-four such Niemeier lattices that exists in twenty-four dimensions, or the single $$\mathrm E_{8}$$ lattice in eight dimensions (these lattices only exist for dimensions $$ d \propto 8$$). 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 $$ f_{\alpha} (r) = e^{-\alpha {r}}$$ of root $$r$$ and $$\alpha > 0$$.

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

In science

 * The atomic number of germanium
 * The freezing point of water at standard atmospheric pressure in degrees Fahrenheit
 * In the Standard Model of particle physics, there are 32 degrees of freedom among the leptons and all bosons that interact with them (including the graviton, which is generally expected to exist, and assuming there are no right-handed neutrinos)

Astronomy

 * Messier 32, a magnitude 9.0 galaxy in the constellation Andromeda which is a companion to M31.
 * The New General Catalogue object NGC 32, a star in the constellation Pegasus

In music

 * A thirty-second note or demisemiquaver is a note played for 1/32 of the duration of a whole note
 * The number of completed, numbered piano sonatas by Ludwig van Beethoven
 * "32 Footsteps", a song by They Might Be Giants
 * "The Chamber of 32 Doors", a song by Genesis, from their 1974 concept album The Lamb Lies Down on Broadway
 * "32", a song on Mr. Mister's debut album I Wear the Face
 * "32", a song by electro-rock group Carpark North
 * "Thirty Two", a song by Van Morrison on the album New York Sessions '67
 * ThirtyTwo is the fourth album by English band Reverend and the Makers
 * "32 Pennies", a song on Warrant's 1989 debut album Dirty Rotten Filthy Stinking Rich
 * The number of rays in the Japanese Rising Sun on the cover of Incubus' 2006 album Light Grenades
 * "32 Ways To Die", a song on Sum41's album Half Hour of Power
 * The shortened pseudonym of UK rapper Wretch 32

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 chess, the total number of black squares on the board, the total number of white squares, and the total number of pieces (black and white) at the beginning of the game.
 * The number of teams in the National Football League.
 * The number of teams in the National Hockey League.
 * In association football:
 * The FIFA World Cup final tournament has featured 32 men's national teams from 1998 through 2022, after which the field will expand to 48.
 * The FIFA Women's World Cup final tournament will feature 32 national teams starting with the next edition in 2023.
 * The ball used in association football is most often made with 32 panels of leather or synthetic material.

In other fields
Thirty-two could also refer to:
 * The number of teeth of a full set of teeth in an adult human, including wisdom teeth
 * The size of a databus in bits: 32-bit
 * The size, in bits, of certain integer data types, used in computer representations of numbers
 * IPv4 uses 32-bit (4-byte) addresses
 * ASCII and Unicode code point for space
 * The code for international direct dial phone calls to Belgium
 * In the title Thirty-Two Short Films About Glenn Gould, starring Colm Feore
 * The Article 32 of the UCMJ concerns pre-trial investigations. Such a hearing is often called an "Article 32 hearing"
 * The caliber .32 ACP
 * The number of the French department Gers
 * The traditional 32 counties of Ireland