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T o g g l e M a t h e m a t i c s s u b s e c t i o n
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2
I n s c i e n c e
T o g g l e I n s c i e n c e s u b s e c t i o n
2 . 1
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4
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T o g g l e t h e t a b l e o f c o n t e n t s
6 3 ( n u m b e r )
7 3 l a n g u a g e s
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
63 (sixty-three ) is the natural number following 62 and preceding 64 .
Mathematics
[ edit ]
63 is the sum of the first six powers of 2 (2 0 + 21 + ... 2 5 ). 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
p
2
×
q
{\displaystyle \,p^{2}\times q}
and the second of the form
3
2
×
q
{\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
a
>
b
>
0
{\displaystyle a>b>0}
are coprime integers for any integer
n
≥
1
{\displaystyle n\geq 1}
, there exists a primitive prime divisor
p
{\displaystyle p}
that divides
a
n
−
b
n
{\displaystyle a^{n}-b^{n}}
and does not divide
a
k
−
b
k
{\displaystyle a^{k}-b^{k}}
for any positive integer
k
<
n
{\displaystyle k<n}
, except for when
n
=
1
{\displaystyle n=1}
,
a
−
b
=
1
;
{\displaystyle a-b=1;\;}
with
a
n
−
b
n
=
1
{\displaystyle a^{n}-b^{n}=1}
having no prime divisors,
n
=
2
{\displaystyle n=2}
,
a
+
b
{\displaystyle a+b\;}
a power of two , where any odd prime factors of
a
2
−
b
2
=
(
a
+
b
)
(
a
1
−
b
1
)
{\displaystyle a^{2}-b^{2}=(a+b)(a^{1}-b^{1})}
are contained in
a
1
−
b
1
{\displaystyle a^{1}-b^{1}}
, which is even ;
and for a special case where
n
=
6
{\displaystyle n=6}
with
a
=
2
{\displaystyle a=2}
and
b
=
1
{\displaystyle b=1}
, which yields
a
6
−
b
6
=
2
6
−
1
6
=
63
=
3
2
×
7
=
(
a
2
−
b
2
)
2
(
a
3
−
b
3
)
{\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
2
n
−
1
{\displaystyle 2^{n}-1}
with an
n
{\displaystyle n}
of
6
{\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
2
n
+
1
{\displaystyle 2n+1}
, is 63.[8] It is also the fourth Woodall number of the form
n
⋅
2
n
−
1
{\displaystyle n\cdot 2^{n}-1}
with
n
=
4
{\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
{
1
,
2
,
3
,
5
,
6
,
7
,
10
,
14
,
15
}
{\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
3
×
3
{\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
[ edit ]
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
q
=
p
n
{\displaystyle q=p^{n}}
(with
p
{\displaystyle p}
prime and
n
{\displaystyle n}
a positive integer) by the GCD of
(
a
,
b
)
{\displaystyle (a,b)}
, and a
∏
{\displaystyle \textstyle \prod }
(in capital pi notation , product over a set of
i
{\displaystyle i}
terms):[14]
q
63
(
2
,
q
−
1
)
∏
i
∈
{
2
,
6
,
8
,
10
,
12
,
14
,
18
}
(
q
i
−
1
)
,
{\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,
E
7
(
q
)
.
{\displaystyle E_{7}(q ).}
q
36
(
3
,
q
−
1
)
∏
i
∈
{
2
,
5
,
6
,
8
,
9
,
12
}
(
q
i
−
1
)
,
{\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,
E
6
(
q
)
.
{\displaystyle E_{6}(q ).}
q
36
(
3
,
q
+
1
)
∏
i
∈
{
2
,
5
,
6
,
8
,
9
,
12
}
(
q
i
−
(
−
1
)
i
)
,
{\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 ,
2
E
6
(
q
2
)
.
{\displaystyle ^{2}E_{6}(q^{2}).}
Lie algebra
E
6
{\displaystyle E_{6}}
holds thirty-six positive roots in sixth-dimensional space, while
E
7
{\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
W
(
E
7
)
{\displaystyle W(E_{7})}
, with order
2
63
{\displaystyle 2^{63}}
where the previous
W
(
E
6
)
{\displaystyle W(E_{6})}
has order
2
36
{\displaystyle 2^{36}}
; these are associated, respectively, with
E
7
{\displaystyle E_{7}}
and
E
6
.
{\displaystyle E_{6}.}
[16]
There are 63 uniform polytopes in the sixth dimension that are generated from the abstract hypercubic
B
6
{\displaystyle \mathrm {B_{6}} }
Coxeter group (sometimes, the demicube is also included in this family),[17] that is associated with classical Chevalley Lie algebra
B
6
{\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
A
6
{\displaystyle \mathrm {A_{6}} }
simplex Coxeter group, when counting self-dual configurations of the regular 6-simplex separately.[17] In similar fashion,
A
6
{\displaystyle \mathrm {A_{6}} }
is associated with classical Chevalley Lie algebra
A
6
{\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
I
h
{\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,
I
h
{\displaystyle \mathrm {I_{h}} }
of order 120 contains a total of thirty-one axes of symmetry ;[19] specifically, the
E
8
{\displaystyle \mathbb {E_{8}} }
lattice that is associated with exceptional Lie algebra
E
8
{\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,
σ
(
63
)
=
104
{\displaystyle \sigma (63 )=104}
,[22] is equal to the constant term
a
(
0
)
=
104
{\displaystyle a(0)=104}
that belongs to the principal modular function (McKay–Thompson series )
T
2
A
(
τ
)
{\displaystyle T_{2A}(\tau )}
of sporadic group
B
{\displaystyle \mathrm {B} }
, the second largest such group after the Friendly Giant
F
1
{\displaystyle \mathrm {F} _{1}}
.[23] This value is also the value of the minimal faithful dimensional representation of the Tits group
T
{\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
F
4
{\displaystyle F_{4}}
, in 52 dimensions.
In science
[ edit ]
Astronomy
[ edit ]
In other fields
[ edit ]
Sixty-three is also:
In religion
[ edit ]
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
[ edit ]
^ 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 .
^ 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 .
^ 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 .
^ a b 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 .
^ 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 .
^ 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 .
^ 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 .
^ Sloane, N. J. A. (ed.). "Sequence A003261 (Woodall numbers)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ 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 .
^ 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 .
^ Sloane, N. J. A. (ed.). "Sequence A001850 (Central Delannoy numbers)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ 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 .
^ 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 .
^ 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 .
^ Sekiguchi, Jiro (2023). "Simple singularity of type E7 and the complex reflection group ST34". arXiv :2311.16629 [math.AG ]. Bibcode :2023arXiv231116629S .
^ a b 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 .
^ Webb, Robert. "Enumeration of Stellations" . Stella . Archived from the original on 2022-11-26. Retrieved 2023-09-21 .
^ 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 .{{cite book }}
: CS1 maint: location missing publisher (link )
^ 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 .
^ 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 T 1 , O1 and I1 , respectively).
^ 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 .
^ 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 .
j
2
A
(
τ
)
=
T
2
A
(
τ
)
+
104
=
1
q
+
104
+
4372
q
+
96256
q
2
+
⋯
{\displaystyle j_{2A}(\tau )=T_{2A}(\tau )+104={\frac {1}{q}}+104+4372q+96256q^{2}+\cdots }
^ 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 .
t
e
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=63_(number)&oldid=1235720283 "
C a t e g o r y :
● I n t e g e r s
H i d d e n c a t e g o r i e s :
● C S 1 m a i n t : l o c a t i o n m i s s i n g p u b l i s h e r
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 2 0 J u l y 2 0 2 4 , a t 2 1 : 2 2 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w