63 (number)
Template:Lead too short Template:Infobox number63 (sixty-three) is the natural number following 62 and preceding 64. It is a composite number.
Mathematics
63 is the sum of the first six powers of 2 (20 + 21 + ... 25). It is the eighth highly cototient number,<ref>Template:Cite OEIS</ref> and the fourth centered octahedral number after 7 and 25.<ref>Template:Cite OEIS</ref> For five unlabeled elements, there are 63 posets.<ref>Template:Cite OEIS</ref>
Sixty-three is the seventh square-prime of the form <math>\, p^{2} \times q</math> and the second of the form <math>3^{2} \times q</math>. 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.
Zsigmondy's theorem states that where <math>a>b>0</math> are coprime integers for any integer <math>n \ge 1</math>, there exists a primitive prime divisor <math>p</math> that divides <math>a^n-b^n</math> and does not divide <math>a^k-b^k</math> for any positive integer <math>k<n</math>, except for when
- <math>n=1</math>, <math>a-b=1; \;</math> with <math>a^n-b^n=1</math> having no prime divisors,
- <math>n=2</math>, <math>a+b \;</math> a power of two, where any odd prime factors of <math>a^2-b^2=(a+b)(a^1-b^1)</math> are contained in <math>a^1-b^1</math>, which is even;
and for a special case where <math>n=6</math> with <math>a=2</math> and <math>b=1</math>, which yields <math>a^6-b^6=2^6-1^6=63=3^2\times 7=(a^2-b^2)^2 (a^3-b^3)</math>.<ref>Template:Cite book</ref>
63 is a Mersenne number of the form <math>2^{n} - 1 </math> with an <math>n</math> of <math>6</math>,<ref name=MersN>Template:Cite OEIS</ref> however this does not yield a Mersenne prime, as 63 is the forty-fourth composite number.<ref>Template:Cite OEIS</ref> 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).<ref>Template:Cite OEIS</ref> In the list of Mersenne numbers, 63 lies between Mersenne primes 31 and 127, with 127 the thirty-first prime number.<ref name=MersN /> The thirty-first odd number, of the simplest form <math>2n+1</math>, is 63.<ref>Template:Cite OEIS</ref> It is also the fourth Woodall number of the form <math>n \cdot 2^n - 1</math> with <math>n = 4</math>, with the previous members being 1, 7 and 23 (they add to 31, the third Mersenne prime).<ref>Template:Cite OEIS</ref>
In the integer positive definite quadratic matrix <math>\{1, 2, 3, 5, 6, 7, 10, 14, 15\}</math> representative of all (even and odd) integers,<ref>Template:Cite OEIS</ref><ref>Template:Cite book</ref> the sum of all nine terms is equal to 63.
63 is the third Delannoy number, which represents the number of pathways in a <math>3 \times 3</math> grid from a southwest corner to a northeast corner, using only single steps northward, eastward, or northeasterly.<ref>Template:Cite OEIS</ref>
Finite simple groups
63 holds thirty-six integers that are relatively prime with itself (and up to), equivalently its Euler totient.<ref>Template:Cite OEIS</ref> 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 <math>q = p^{n}</math> (with <math>p</math> prime and <math>n</math> a positive integer) by the GCD of <math>(a, b)</math>, and a <math>\textstyle \prod</math> (in capital pi notation, product over a set of <math>i</math> terms):<ref name="Gallian">Template:Cite journal</ref>
- <math>\frac{q^{63}}{(2, q - 1)}\prod_{i\in\{2, 6, 8, 10, 12, 14, 18\} } \left(q^i - 1\right),</math> the order of exceptional Chevalley finite simple group of Lie type, <math>E_{7}(q).</math>
- <math>\frac{q^{36}}{(3, q - 1)}\prod_{i\in\{2, 5, 6, 8, 9, 12\} } \left(q^i - 1\right),</math> the order of exceptional Chevalley finite simple group of Lie type, <math>E_{6}(q).</math>
- <math>\frac{q^{36}}{(3, q + 1)}\prod_{i\in\{2, 5, 6, 8, 9, 12\} } \left(q^i - (-1)^i\right),</math> the order of one of two exceptional Steinberg groups, <math>^{2}E_{6}(q^{2}).</math>
Lie algebra <math>E_{6}</math> holds thirty-six positive roots in sixth-dimensional space, while <math>E_{7}</math> holds sixty-three positive root vectors in the seven-dimensional space (with one hundred and twenty-six total root vectors, twice 63).<ref>Template:Cite book</ref> The thirty-sixth-largest of thirty-seven total complex reflection groups is <math>W(E_{7})</math>, with order <math>2^{63}</math> where the previous <math>W(E_{6})</math> has order <math>2^{36}</math>; these are associated, respectively, with <math>E_{7}</math> and <math>E_{6}.</math><ref>Template:Cite arXiv Template:Bibcode.</ref>
There are 63 uniform polytopes in the sixth dimension that are generated from the abstract hypercubic <math>\mathrm {B_{6}}</math> Coxeter group (sometimes, the demicube is also included in this family),<ref name=Coxeter>Template:Cite journal</ref> that is associated with classical Chevalley Lie algebra <math>B_{6}</math> 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 <math>\mathrm {A_{6}}</math> simplex Coxeter group, when counting self-dual configurations of the regular 6-simplex separately.<ref name=Coxeter /> In similar fashion, <math>\mathrm {A_{6}}</math> is associated with classical Chevalley Lie algebra <math>A_{6}</math> 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 <math>\mathrm {I_{h}}</math>, 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).<ref>Template:Cite web</ref> 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, <math>\mathrm {I_{h}}</math> of order 120 contains a total of thirty-one axes of symmetry;<ref>Template:Cite book</ref> specifically, the <math>\mathbb {E_{8}}</math> lattice that is associated with exceptional Lie algebra <math>{E_{8}}</math> contains symmetries that can be traced back to the regular icosahedron via the icosians.<ref>Template:Cite journal</ref> 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.<ref>Template:Cite journal
- See Tables 5, 6 and 7 (groups T1, O1 and I1, respectively).</ref>
Otherwise, the sum of the divisors of sixty-three, <math>\sigma(63)=104</math>,<ref>Template:Cite OEIS</ref> is equal to the constant term <math>a(0) = 104</math> that belongs to the principal modular function (McKay–Thompson series) <math>T_{2A}(\tau)</math> of sporadic group <math>\mathrm B</math>, the second largest such group after the Friendly Giant <math>\mathrm F_{1}</math>.<ref>Template:Cite OEIS
- <math>j_{2A}(\tau) = T_{2A}(\tau)+104 = \frac{1}{q} + 104 + 4372q + 96256q^2 + \cdots</math></ref> This value is also the value of the minimal faithful dimensional representation of the Tits group <math>\mathrm T</math>,<ref>Template:Cite journal</ref> 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 <math>F_{4}</math>, in 52 dimensions.
References
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