Natural number
32 (thirty-two ) is thenatural number following31 and preceding33 .
32 is the fifthpower of two (2 5 {\displaystyle 2^{5}} fifth-power of the formp 5 {\displaystyle p^{5}} p {\displaystyle p} totient summatory function Φ ( n ) {\displaystyle \Phi (n)} 10 integers,[ 1] n {\displaystyle n} 7 solutions forφ ( n ) {\displaystyle \varphi (n)}
Thealiquot sum of a power of two is always one less than the number itself, therefore the aliquot sum of 32 is31 .[ 2]
32 = 1 1 + 2 2 + 3 3 32 = ( 1 × 4 ) + ( 2 × 5 ) + ( 3 × 6 ) 32 = ( 1 × 2 ) + ( 1 × 2 × 3 ) + ( 1 × 2 × 3 × 4 ) 32 = ( 1 × 2 × 3 ) + ( 4 × 5 ) + ( 6 ) {\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)\\32&=(1\times 2\times 3)+(4\times 5)+(6)\end{aligned}}}
The product betweenneighbor numbers of23 permutation of thedigits of 32 indecimal , is equal to the sum of the first 32integers :22 × 24 = 528 {\displaystyle 22\times 24=528} [ 3] [ a]
32 is also aLeyland number expressible in the formx y + y x {\displaystyle x^{y}+y^{x}} [ 5] [ b] 32 = 2 4 + 4 2 . {\displaystyle 32=2^{4}+4^{2}.}
The eleventh Mersenne number is the first to have a primeexponent (11 ) that does not yield aMersenne prime , equal to:[ 7] [ c] 2047 = 32 2 + ( 31 × 33 ) = 1024 + 1023 = 2 11 − 1. {\displaystyle 2047=32^{2}+(31\times 33)=1024+1023=2^{11}-1.}
When read inbinary , the first 32 rows ofPascal's Triangle represent the thirty-two divisors that belong to the largestconstructible polygon . The product of the five knownFermat primes is equal to the number of sides of the largest regularconstructible polygon with astraightedge andcompass that has an odd number of sides, with a total ofsides numbering2 32 − 1 = 3 ⋅ 5 ⋅ 17 ⋅ 257 ⋅ 65 537 = 4 294 967 295. {\displaystyle 2^{32}-1=3\cdot 5\cdot 17\cdot 257\cdot 65\;537=4\;294\;967\;295.}
The first 32 rows ofPascal's triangle read as singlebinary numbers represent the 32divisors that belong to this number, which is also the number of sides of all odd-sided constructible polygons with simple tools alone (if themonogon is also included).[ 10]
There are also a total of 32uniform colorings to the 11regular andsemiregular tilings .[ 11]
There are 32 three-dimensionalcrystallographic point groups [ 12] crystal families ,[ 13] maximum determinant in a 7 by 7 matrix of only zeroes and ones is 32.[ 14] sedenions generate a non-commutative loop S L {\displaystyle \mathbb {S} _{L}} order 32,[ 15] dimensions , there are at least 1,160,000,000even unimodular lattices (ofdeterminants 1 or −1);[ 16] Niemeier lattices that exists in twenty-four dimensions, or the singleE 8 {\displaystyle \mathrm {E} _{8}} d ∝ 8 {\displaystyle d\propto 8} non-critical even unimodular lattices that do not interact with aGaussian potential function of the formf α ( r ) = e − α r {\displaystyle f_{\alpha }(r)=e^{-\alpha {r}}} r {\displaystyle r} α > 0 {\displaystyle \alpha >0} [ 17]
32 is the furthest point in the set ofnatural numbers N 0 {\displaystyle \mathbb {N} _{0}} primes (2, 3, 5, ..., 31) to non-primes (0, 1, 4, ..., 32) is1 2 . {\displaystyle {\tfrac {1}{2}}.} [ d]
Thetrigintaduonions form a 32-dimensionalhypercomplex number system.[ 20]
^ 32 is the ninth10 -happy number , while 23 is the sixth.[ 4] 55 , which is the tenthtriangular number ,[ 3] 9 = 3 2 . {\displaystyle 9=3^{2}.} ^ On the other hand, aregular 32-sidedtriacontadigon contains2 3 + 3 2 = 17 {\displaystyle 2^{3}+3^{2}=17} symmetries .[ 6] hexadecagon contains 14 symmetries, an 8-sidedoctagon contains 11 symmetries, and asquare contains 8 symmetries. ^ Specifically, 31 is the eleventh prime number, equal to the sum of 20 and its compositeindex 11, where 33 is the twenty-first composite number, equal to the sum of 21 and its composite index 12 (which arepalindromic 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, whichmaps to 31 through its prime index of 11, and 33 by afactor of 11, that is the composite index of 20; the aliquot part of 32 is 31 as well).[ 2] prime number theorem . ^ 29 is the only earlier point, where there are twenty non primes, and ten primes.40 8 th pair of sexy primes (37, 43),[ 18] composite numbers (up to) is1 / 2 . Where 68 is the forty-eighth composite, 48 is the thirty second, with the difference68 –48 =20 [ 8] N + {\displaystyle \mathbb {N} ^{+}} 11 × 33 =363 represents thethirty-second 0 Mertens function M (n )[ 19] ^ Sloane, N. J. A. (ed.)."Sequence A002088 (Sum of totient function)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-05-04 .^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.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-10 .^a b Sloane, N. J. A. (ed.)."Sequence A000217 (Triangular numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-05-04 .^ "Sloane's A007770 : Happy numbers" .The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2016-05-31 .^ "Sloane's A076980 : Leyland numbers" .The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2016-05-31 .^ Conway, John H. ; Burgiel, Heidi;Goodman-Strauss, Chaim (2008). "Chapter 20: Generalized Schaefli symbols (Types of symmetry of a polygon)".The Symmetries of Things CRC Press (Taylor & Francis ). pp. 275– 277.doi :10.1201/b21368 .ISBN 978-1-56881-220-5 OCLC 181862605 .Zbl 1173.00001 .^ 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.))" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-08 .^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.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-08 .^ Sloane, N. J. A. (ed.)."Sequence A00040 (The prime numbers.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-08 .^ 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 .^ Grünbaum, Branko ;Shephard, G. C. (1987). "Section 2.9 Archimedean and uniform colorings".Tilings and Patterns 102– 107.doi :10.2307/2323457 .ISBN 0-7167-1193-1 JSTOR 2323457 .OCLC 13092426 .S2CID 119730123 .^ Sloane, N. J. A. (ed.)."Sequence A004028 (Number of geometric n-dimensional crystal classes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-11-08 .^ Sloane, N. J. A. (ed.)."Sequence A004032 (Number of n-dimensional crystal families.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-11-08 .^ Sloane, N. J. A. (ed.)."Sequence A003432 (Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-04-04 .^ 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. : TheMathematical Society of the Philippines :13– 15.ISSN 0115-6926 .Zbl 1155.20315 . ^ Baez, John C. (November 15, 2014)."Integral Octonions (Part 8)" .John Baez's Stuff .U.C. Riverside , Department of Mathematics. Retrieved2023-05-04 .^ 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 . ^ Sloane, N. J. A. (ed.)."Sequence A156274 (List of prime pairs of the form (p, p+6).)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-11 .^ Sloane, N. J. A. (ed.)."Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-11 .^ 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 .
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