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{{wiktionary|32|thirty-two|XXXII}}
{{Infobox number
| number =  32
| divisor = 1, 2, 4, 8, 16, 32
|lang1=[[ASCII|ASCII value]]
|lang1 symbol=[[Space (punctuation)|[space]]]
}}
'''32''' ('''thirty-two''') is the [[natural number]] [[following]] [[31 (number)|31]] and preceding [[33 (number)|33]].

== Mathematics ==
'''32''' is the fifth [[power of two]] (<math>2^{5}</math>), making it the first non-unitary [[Fifth power (algebra)|fifth-power]] of the form <math>p^{5}</math> where <math>p</math> is prime. 32 is the [[totient summatory function]] <math>\Phi(n)</math> over the first [[10]] integers,<ref>{{Cite OEIS |A002088 |Sum of totient function |access-date=2023-05-04 }}</ref> and the smallest number <math>n</math> with exactly [[7]] solutions for <math>\varphi (n)</math>. 

The [[aliquot sum]] of a power of two is always one less than the number itself, therefore the aliquot sum of 32 is [[31 (number)|31]].<ref name="AliSum">{{Cite OEIS |A001065 |Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. |access-date=2024-01-10 }}</ref>

<math display=block>
\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) \\
32 & = (1\times2\times3)+(4\times5)+(6)
\end{align}</math>

The product between ''neighbor'' numbers of '''[[23 (number)|23]]''', the dual [[permutation]] of the [[Numerical digit|digits]] of 32 in [[Base ten|decimal]], is equal to the sum of the first 32 [[integer]]s: <math>22 \times 24= 528</math>.<ref name="triangular">{{Cite OEIS |A000217 |Triangular numbers |access-date=2023-05-04 }}</ref>{{efn|1=32 is the ninth {{math|10}}-[[happy number]], while 23 is the sixth.<ref>{{Cite web|url=https://oeis.org/A007770|title=Sloane's A007770 : Happy numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}</ref> Their sum is [[55 (number)|55]], which is the tenth [[triangular number]],<ref name="triangular" /> while their difference is <math>9 = 3^{2}.</math> }}

32 is also a [[Leyland number]] expressible in the form <math>x^y + y^x</math>, where:<ref>{{Cite web|url=https://oeis.org/A076980|title=Sloane's A076980 : Leyland numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}</ref>{{efn|1=On the other hand, a [[Regular polygon|regular]] 32-sided [[triacontadigon]] contains <math>2^{3} + 3^{2} = 17</math> distinct [[Symmetry (geometry)|symmetries]].<ref>{{Cite book |last1=Conway |first1=John H. |author1-link=John Horton Conway|last2=Burgiel |first2=Heidi |last3=Goodman-Strauss |first3=Chaim |author3-link=Chaim Goodman-Strauss |title=The Symmetries of Things |url=https://www.taylorfrancis.com/books/mono/10.1201/b21368/symmetries-things-chaim-goodman-strauss-john-conway-heidi-burgiel |chapter=Chapter 20: Generalized Schaefli symbols (Types of symmetry of a polygon) |publisher=[[CRC Press]] ([[Taylor & Francis]]) |edition=1st |year=2008 |location=New York |pages=275–277 |doi=10.1201/b21368  |isbn= 978-1-56881-220-5 |oclc=181862605 |zbl=1173.00001 }}</ref><br />For comparison, a 16-sided [[hexadecagon]] contains 14 symmetries, an 8-sided [[octagon]] contains 11 symmetries, and a [[square]] contains 8 symmetries. }}
<math display=block>32=2^{4} + 4^{2}.</math>

The eleventh Mersenne number is the first to have a prime [[exponent]] ([[11 (number)|11]]) that does not yield a [[Mersenne prime]], equal to:<ref>{{Cite OEIS |A000225 |a(n) equal to 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.) |access-date=2024-01-08 }}</ref>{{efn|1=Specifically, 31 is the eleventh prime number, equal to the sum of 20 and its composite [[Sequence|index]] 11, where 33 is the twenty-first composite number, equal to the sum of 21 and its composite index 12 (which are [[palindromic number]]s).<ref name="CompN">{{Cite OEIS |A002808 |The composite numbers: numbers n of the form x*y for x > 1 and y > 1. |access-date=2024-01-08 }}</ref><ref>{{Cite OEIS |A00040 |The prime numbers. |access-date=2024-01-08 }}</ref> '''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 [[Map (mathematics)|maps]] to 31 through its prime index of 11, and 33 by a [[Factor (arithmetic)|factor]] of 11, that is the composite index of 20; the aliquot part of 32 is 31 as well).<ref name="AliSum" /> This is due to the fact that the ratio of composites to primes increases very rapidly, by the [[prime number theorem]]. }}
<math display=block>2047 = 32^{2} + (31 \times 33) = 1024 + 1023 = 2^{11} - 1.</math>

[[File:Binary Pascal's Triangle.png|250px|right|thumb|When read in [[Binary number|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 prime]]s is equal to the number of sides of the largest regular [[constructible polygon]] with a [[Ruler|straightedge]] and [[Compass (drawing tool)|compass]] that has an odd number of sides, with a total of [[Edge (geometry)|sides]] numbering 
<math display=block>2^{32} - 1 = 3\cdot5\cdot17\cdot257\cdot65\;537 = 4\;294\;967\;295.</math>

The first 32 rows of [[Pascal's triangle]] read as single [[Binary number|binary numbers]] represent the 32 [[divisor]]s 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).<ref>{{Cite book |last1=Conway |first1=John H. |author1-link= John Horton Conway |last2=Guy |first2=Richard K. |author-link2=Richard K. Guy |title=The Book of Numbers |chapter=The Primacy of Primes |year=1996 |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4612-4072-3_5 |publisher=Copernicus ([[Springer Publishing|Springer]]) |location=New York, NY |pages=137–142 |isbn=978-1-4612-8488-8 |doi=10.1007/978-1-4612-4072-3 |oclc=32854557 |s2cid=115239655 }}</ref>

There are also a total of 32 [[List of Euclidean uniform tilings#Uniform colorings|uniform coloring]]s to the 11 [[Euclidean tilings by convex regular polygons#Regular tilings|regular]] and [[Euclidean tilings by convex regular polygons#Archimedean, uniform or semiregular tilings|semiregular]] [[List of Euclidean uniform tilings|tilings]].<ref>{{Cite book |author1=Grünbaum, Branko |author-link=Branko Grünbaum |author2=Shephard, G. C. |author2-link=G.C. Shephard |url-access=registration |url=https://archive.org/details/isbn_0716711931 |title=Tilings and Patterns |chapter=Section 2.9 Archimedean and uniform colorings |publisher=W. H. Freeman and Company |location=New York |year=1987 |pages=102–107 |doi=10.2307/2323457 |isbn=0-7167-1193-1 |oclc=13092426 |jstor=2323457 |s2cid=119730123 }}</ref>

There are 32 three-dimensional [[crystallographic point group]]s<ref>{{Cite OEIS |A004028 |Number of geometric n-dimensional crystal classes. |access-date=2022-11-08}}</ref> and 32 five-dimensional [[Crystal system#Classifications|crystal families]],<ref>{{Cite OEIS |A004032 |Number of n-dimensional crystal families. |access-date=2022-11-08 }}</ref> and the [[Hadamard's maximal determinant problem|maximum determinant]] in a 7 by 7 matrix of only zeroes and ones is 32.<ref>{{Cite OEIS |A003432 |Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n. |access-date=2023-04-04 }}</ref>  In sixteen dimensions, the [[sedenion]]s generate a non-[[Commutative property|commutative]] [[Quasigroup#Loops|loop]] <math>\mathbb {S}_{L}</math> of [[Group order|order]] 32,<ref>{{Cite journal |last1=Cawagas |first1=Raoul E. |last2=Gutierrez |first2=Sheree Ann G. |title=The Subloop Structure of the Cayley-Dickson Sedenion Loop |url=http://mathsociety.ph/matimyas/images/vol28/CawagasMatimyas.pdf |journal=Matimyás Matematika |volume=28 |number=1–3  |pages=13–15 |publisher=The [[Mathematical Society of the Philippines]] |location=Diliman, [[Quezon City|Q.C.]] |year=2005 |issn=0115-6926 |doi= |zbl=1155.20315 }}</ref> and in thirty-two [[dimension]]s, there are at least 1,160,000,000 [[Unimodular lattice|even unimodular lattice]]s (of [[Lattice (group)#Dividing space according to a lattice|determinants]] 1 or −1);<ref>{{Cite web |last1=Baez |first = John C. |author-link=John C. Baez |title=Integral Octonions (Part 8) |website=John Baez's Stuff |url=https://math.ucr.edu/home/baez/octonions/integers/integers_8.html |publisher=[[University of California, Riverside|U.C. Riverside]], Department of Mathematics |date=November 15, 2014 |access-date=2023-05-04 }}</ref> which is a marked increase from the twenty-four such [[Niemeier lattice|Niemeier]] lattices that exists in twenty-four dimensions, or the single <math>\mathrm E_{8}</math> lattice in eight dimensions (these lattices only exist for dimensions <math> d \propto 8</math>). Furthermore, the 32nd dimension is the first dimension that holds ''non-critical'' even unimodular lattices that do not interact with a [[Gaussian function|Gaussian potential function]] of the form <math> f_{\alpha} (r) = e^{-\alpha {r}}</math> of root <math>r</math> and <math>\alpha > 0</math>.<ref>{{Cite journal |last1=Heimendahl |first1=Arne |last2=Marafioti |first2=Aurelio |last3=Thiemeyer |first3=Antonia |last4=Vallentin |first4=Frank |last5=Zimmermann |first5=Marc C. |display-authors=2 |title= Critical Even Unimodular Lattices in the Gaussian Core Model |journal=[[International Mathematics Research Notices]] |publisher=[[Oxford University Press]] |volume=1 |issue=6 |page=5352 |location=Oxford |date=June 2022 |doi=10.1093/imrn/rnac164  |arxiv=2105.07868 |s2cid=234742712 |zbl=1159.11020 }}</ref>

32 is the furthest point in the set of [[natural number]]s <math>\mathbb {N}_{0}</math> where the ratio of [[Prime number|primes]] (2, 3, 5, ..., 31) to non-primes (0, 1, 4, ..., 32) is <math>\tfrac {1}{2}.</math>{{efn|1=[[29 (number)|29]] is the only earlier point, where there are twenty non primes, and ten primes. '''[[40 (number)|40]]''' — twice the composite index of 32 — lies between the '''8'''th pair of sexy primes (37, 43),<ref>{{Cite OEIS |A156274 |List of prime pairs of the form (p, p+6). |access-date=2024-01-11 }}</ref> which represent the only two points in the set of natural numbers where the ratio of prime numbers to ''composite numbers'' (up to) is {{sfrac|1|2}}. Where 68 is the forty-eighth composite, 48 is the thirty second, with the difference {{math|1=[[68 (number)|68]] – [[48 (number)|48]] = [[20 (number)|20]] }}, the composite index of 32.<ref name="CompN" /> Otherwise, thirty-two lies midway between primes (23, 41), (17, 47) and (3, 61).<br />At 33, there are 11 numbers that are prime and 22 that are not, when considering instead the set of natural numbers <math>\mathbb{N}^+</math> that does not include 0. The product {{math|1=11 × 33 = '''[[363 (number)|363]]''' }} represents the '''''thirty-second''''' number to return {{math|1=[[0]] }} for the [[Mertens function]] {{math|1=''M''(''n'') }}.<ref>{{Cite OEIS |A028442 |Numbers k such that Mertens's function M(k) (A002321) is zero. |access-date=2024-01-11 }}</ref> }}

The [[trigintaduonion]]s form a 32-dimensional [[hypercomplex number]] system.<ref>{{cite journal | last=Saniga | first=Metod | last2=Holweck | first2=Frédéric | last3=Pracna | first3=Petr | title=From Cayley-Dickson Algebras to Combinatorial Grassmannians | journal=Mathematics | publisher=MDPI AG | volume=3 | issue=4 | date=2015 | issn=2227-7390 | arxiv=1405.6888 | doi=10.3390/math3041192 | doi-access=free | pages=1192–1221}}</ref>

== Notes ==
{{notelist}}

== References ==

{{Reflist}}

== External links ==
* [http://primes.utm.edu/curios/page.php/32.html Prime Curios! 32] from the [[Prime Pages]]

{{Integers|zero}}

{{DEFAULTSORT:32 (Number)}}
[[Category:Integers]]