63 (number)

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Cardinal	sixty-three
Ordinal	63rd (sixty-third)
Factorization	3^2 × 7
Divisors	1, 3, 7, 9, 21, 63
Greek numeral	ΞΓ´
Roman numeral	LXIII
Binary	1111112
Ternary	21003
Senary	1436
Octal	778
Duodecimal	5312
Hexadecimal	3F16

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

Mathematics

63 is the sum of the first six powers of 2 (2⁰ + 2¹ + ... 2⁵). It is the eighth highly cototient number,^[1] and the fourth centered octahedral number after 7 and 25.^[2] For five unlabeled elements, there are 63 posets.^[3]

Sixty-three is the seventh square-prime of the form ${\displaystyle \,p^{2}\times q}$ and the second of the form ${\displaystyle 3^{2}\times q}$. It contains a prime aliquot sum of 41, the thirteenth indexed prime; and part of the aliquot sequence (63, 41, 1, 0) within the 41-aliquot tree.

63 is the third Delannoy number, for the number of ways to travel from a southwest corner to a northeast corner in a 3 by 3 grid.

Zsigmondy's theorem states that where ${\displaystyle a>b>0}$ are coprime integers for any integer ${\displaystyle n\geq 1}$, there exists a primitive prime divisor ${\displaystyle p}$ that divides ${\displaystyle a^{n}-b^{n}}$ and does not divide ${\displaystyle a^{k}-b^{k}}$ for any positive integer ${\displaystyle k<n}$, except for when

• ${\displaystyle n=1}$, ${\displaystyle a-b=1;\;}$ with ${\displaystyle a^{n}-b^{n}=1}$ having no prime divisors,
• ${\displaystyle n=2}$, ${\displaystyle a+b\;}$ a power of two, where any odd prime factors of ${\displaystyle a^{2}-b^{2}=(a+b)(a^{1}-b^{1})}$ are contained in ${\displaystyle a^{1}-b^{1}}$, which is even;

and for a special case where ${\displaystyle n=6}$ with ${\displaystyle a=2}$ and ${\displaystyle b=1}$, which yields ${\displaystyle a^{6}-b^{6}=2^{6}-1^{6}=63=3^{2}\times 7=(a^{2}-b^{2})^{2}(a^{3}-b^{3})}$.^[4]

63 is a Mersenne number of the form ${\displaystyle 2^{n}-1}$ with an ${\displaystyle n}$ of ${\displaystyle 6}$,^[5] however this does not yield a Mersenne prime, as 63 is the forty-fourth composite number.^[6] It is the only number in the Mersenne sequence whose prime factors are each factors of at least one previous element of the sequence (3 and 7, respectively the first and second Mersenne primes).^[7] In the list of Mersenne numbers, 63 lies between Mersenne primes 31 and 127, with 127 the thirty-first prime number.^[5] The thirty-first odd number, of the simplest form ${\displaystyle 2n+1}$, is 63.^[8] It is also the fourth Woodall number of the form ${\displaystyle n\cdot 2^{n}-1}$ with ${\displaystyle n=4}$, with the previous members being 1, 7 and 23 (they add to 31, the third Mersenne prime).^[9]

In the integer positive definite quadratic matrix ${\displaystyle \{1,2,3,5,6,7,10,14,15\}}$ representative of all (even and odd) integers,^[10][11] the sum of all nine terms is equal to 63.

63 is the third Delannoy number, which represents the number of pathways in a ${\displaystyle 3\times 3}$ grid from a southwest corner to a northeast corner, using only single steps northward, eastward, or northeasterly.^[12]

Finite simple groups

63 holds thirty-six integers that are relatively prime with itself (and up to), equivalently its Euler totient.^[13] In the classification of finite simple groups of Lie type, 63 and 36 are both exponents that figure in the orders of three exceptional groups of Lie type. The orders of these groups are equivalent to the product between the quotient of ${\displaystyle q=p^{n}}$ (with ${\displaystyle p}$ prime and ${\displaystyle n}$ a positive integer) by the GCD of ${\displaystyle (a,b)}$, and a ${\displaystyle \textstyle \prod }$ (in capital pi notation, product over a set of ${\displaystyle i}$ terms):^[14]

${\displaystyle {\frac {q^{63}}{(2,q-1)}}\prod _{i\in \{2,6,8,10,12,14,18\}}\left(q^{i}-1\right),}$ the order of exceptional Chevalley finite simple group of Lie type, ${\displaystyle E_{7}(q).}$

${\displaystyle {\frac {q^{36}}{(3,q-1)}}\prod _{i\in \{2,5,6,8,9,12\}}\left(q^{i}-1\right),}$ the order of exceptional Chevalley finite simple group of Lie type, ${\displaystyle E_{6}(q).}$

${\displaystyle {\frac {q^{36}}{(3,q+1)}}\prod _{i\in \{2,5,6,8,9,12\}}\left(q^{i}-(-1)^{i}\right),}$ the order of one of two exceptional Steinberg groups, ${\displaystyle ^{2}E_{6}(q^{2}).}$

Lie algebra ${\displaystyle E_{6}}$ holds thirty-six positive roots in sixth-dimensional space, while ${\displaystyle E_{7}}$ holds sixty-three positive root vectors in the seven-dimensional space (with one hundred and twenty-six total root vectors, twice 63).^[15] The thirty-sixth-largest of thirty-seven total complex reflection groups is ${\displaystyle W(E_{7})}$, with order ${\displaystyle 2^{63}}$ where the previous ${\displaystyle W(E_{6})}$ has order ${\displaystyle 2^{36}}$; these are associated, respectively, with ${\displaystyle E_{7}}$ and ${\displaystyle E_{6}.}$^[16]

There are 63 uniform polytopes in the sixth dimension that are generated from the abstract hypercubic ${\displaystyle \mathrm {B_{6}} }$ Coxeter group (sometimes, the demicube is also included in this family),^[17] that is associated with classical Chevalley Lie algebra ${\displaystyle B_{6}}$ via the orthogonal group and its corresponding special orthogonal Lie algebra (by symmetries shared between unordered and ordered Dynkin diagrams). There are also 36 uniform 6-polytopes that are generated from the ${\displaystyle \mathrm {A_{6}} }$ simplex Coxeter group, when counting self-dual configurations of the regular 6-simplex separately.^[17] In similar fashion, ${\displaystyle \mathrm {A_{6}} }$ is associated with classical Chevalley Lie algebra ${\displaystyle A_{6}}$ through the special linear group and its corresponding special linear Lie algebra.

In the third dimension, there are a total of sixty-three stellations generated with icosahedral symmetry ${\displaystyle \mathrm {I_{h}} }$, using Miller's rules; fifty-nine of these are generated by the regular icosahedron and four by the regular dodecahedron, inclusive (as zeroth indexed stellations for regular figures).^[18] Though the regular tetrahedron and cube do not produce any stellations, the only stellation of the regular octahedron as a stella octangula is a compound of two self-dual tetrahedra that facets the cube, since it shares its vertex arrangement. Overall, ${\displaystyle \mathrm {I_{h}} }$ of order 120 contains a total of thirty-one axes of symmetry;^[19] specifically, the ${\displaystyle \mathbb {E_{8}} }$ lattice that is associated with exceptional Lie algebra ${\displaystyle {E_{8}}}$ contains symmetries that can be traced back to the regular icosahedron via the icosians.^[20] The icosahedron and dodecahedron can inscribe any of the other three Platonic solids, which are all collectively responsible for generating a maximum of thirty-six polyhedra which are either regular (Platonic), semi-regular (Archimedean), or duals to semi-regular polyhedra containing regular vertex-figures (Catalan), when including four enantiomorphs from two semi-regular snub polyhedra and their duals as well as self-dual forms of the tetrahedron.^[21]

Otherwise, the sum of the divisors of sixty-three, ${\displaystyle \sigma (63)=104}$,^[22] is equal to the constant term ${\displaystyle a(0)=104}$ that belongs to the principal modular function (McKay–Thompson series) ${\displaystyle T_{2A}(\tau )}$ of sporadic group ${\displaystyle \mathrm {B} }$, the second largest such group after the Friendly Giant ${\displaystyle \mathrm {F} _{1}}$.^[23] This value is also the value of the minimal faithful dimensional representation of the Tits group ${\displaystyle \mathrm {T} }$,^[24] the only finite simple group that can categorize as being non-strict of Lie type, or loosely sporadic; that is also twice the faithful dimensional representation of exceptional Lie algebra ${\displaystyle F_{4}}$, in 52 dimensions.

In science

• The atomic number of europium.

Astronomy

• Messier object M63, a magnitude 8.5 galaxy in the constellation Canes Venatici, also known as the Sunflower Galaxy.
• The New General Catalogue object NGC 63, a spiral galaxy in the constellation Pisces
• 63 Ausonia, an approximately 100 kilometer in diameter asteroid in the asteroid belt.

In other fields

Sixty-three is also:

• The code for international direct dial calls to the Philippines
• The hull number of the U.S. Navy's aircraft carrier USS Kitty Hawk (CV-63) and the USS Missouri (BB-63)
• The number of the French department Puy-de-Dôme
• The number of groats in a guinea in British pre-decimal currency
• A card game, popular in Carleton County, New Brunswick
• The Stoner 63, a machine gun
• The number of chromosomes found in the offspring of a donkey and a horse
• Class of '63 was a TV movie starring James Brolin (1973)

In religion

• There are 63 Tractates in the Mishna, the compilation of Jewish Law.
• There are 63 Saints (popularly known as Nayanmars) in South Indian Shaivism, particularly in Tamil Nadu, India.
• There are 63 Salakapurusas (great beings) in Jain cosmology.

References

1. Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers: records for a(n) in A063741.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
2. Sloane, N. J. A. (ed.). "Sequence A001845 (Centered octahedral numbers (crystal ball sequence for cubic lattice))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-02.
3. Sloane, N. J. A. (ed.). "Sequence A000112 (Number of partially ordered sets (posets) with n unlabeled elements)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
4. Ribenboim, Paulo (2004). The Little Book of Big Primes (2nd ed.). New York, NY: Springer. p. 27. doi:10.1007/b97621. ISBN 978-0-387-20169-6. OCLC 53223720. S2CID 117794601. Zbl 1087.11001.
5. 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 2023-08-06.
6. Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbersnumbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
7. Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
8. Sloane, N. J. A. (ed.). "Sequence A005408 (The odd numbers: a(n) equal to 2*n + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
9. Sloane, N. J. A. (ed.). "Sequence A003261 (Woodall numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
10. Sloane, N. J. A. (ed.). "Sequence A030050 (Numbers from the Conway-Schneeberger 15-theorem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.
11. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. Vol. 239 (1st ed.). Springer. pp. 312–314. doi:10.1007/978-0-387-49923-9. ISBN 978-0-387-49922-2. OCLC 493636622. Zbl 1119.11001.
12. Sloane, N. J. A. (ed.). "Sequence A001850 (Central Delannoy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
13. Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
14. Gallian, Joseph A. (1976). "The Search for Finite Simple Groups". Mathematics Magazine. 49 (4). Oxfordshire, UK: Taylor & Francis: 174. doi:10.1080/0025570X.1976.11976571. JSTOR 2690115. MR 0414688. S2CID 125460079.
15. Carter, Roger W. (1972). Simple groups of Lie type. Pure and Applied Mathematics (A Series of Texts and Monographs). Vol. XXXVIII (1st ed.). Wiley-Interscience. p. 43. ISBN 978-0471506836. OCLC 609240. Zbl 0248.20015.
16. Sekiguchi, Jiro (2023). "Simple singularity of type E₇ and the complex reflection group ST34". arXiv:2311.16629 [math.AG]. Bibcode:2023arXiv231116629S.
17. Coxeter, H.S.M. (1988). "Regular and Semi-Regular Polytopes. III". Mathematische Zeitschrift. 200. Berlin: Springer-Verlag: 4–7. doi:10.1007/BF01161745. S2CID 186237142. Zbl 0633.52006.
18. Webb, Robert. "Enumeration of Stellations". Stella. Archived from the original on 2022-11-26. Retrieved 2023-09-21.
19. Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Proceedings of the Bridges Conference. Winfield, Kansas. p. 196. ISBN 978-0966520101. OCLC 59580549. S2CID 202679388.
20. Baez, John C. (2018). "From the Icosahedron to E₈". London Mathematical Society Newsletter. 476: 18–23. arXiv:1712.06436. MR 3792329. S2CID 119151549. Zbl 1476.51020.
21. Har’El, Zvi (1993). "Uniform Solution for Uniform Polyhedra" (PDF). Geometriae Dedicata. 47. Netherlands: Springer Publishing: 57–110. doi:10.1007/BF01263494. MR 1230107. S2CID 120995279. Zbl 0784.51020.

    See Tables 5, 6 and 7 (groups T₁, O₁ and I₁, respectively).

22. Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) equal to sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-06.
23. Sloane, N. J. A. (ed.). "Sequence A007267 (Expansion of 16 * (1 + k^2)^4 /(k * k'^2)^2 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-31.

    ${\displaystyle j_{2A}(\tau )=T_{2A}(\tau )+104={\frac {1}{q}}+104+4372q+96256q^{2}+\cdots }$

24. Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type". Communications in Algebra. 29 (5). Philadelphia, PA: Taylor & Francis: 2151. doi:10.1081/AGB-100002175. MR 1837968. S2CID 122060727. Zbl 1004.20003.