Natural number
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
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
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
Astronomy
In other fields
Sixty-three is also:
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
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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).
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MR
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S2CID
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Zbl
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100,000
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