63 (number)

← 62 63 64 →
→
Cardinal	sixty-three
Ordinal	63rd (sixty-third)
Factorization	32 × 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

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.

Zsigmondy's theorem states that where ${\displaystyle a>b>0}$ are

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 ${\displaystyle 6}$,

, 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

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 ${\displaystyle \mathrm {B_{6}} }$

${\displaystyle B_{6}}$ via the ). There are also 36 uniform 6-polytopes that are generated from the ${\displaystyle \mathrm {A_{6}} }$ 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

. 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

) ${\displaystyle T_{2A}(\tau )}$ 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 ${\displaystyle \mathrm {T} }$,

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

• Sunflower Galaxy
  .
• The New General Catalogue object NGC 63, a spiral galaxy in the constellation Pisces

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
  British pre-decimal currency
• 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. Zbl 1087.11001
   .
5. ^ ^{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.
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. 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. S2CID 125460079
    .
15. Zbl 0248.20015
    .
16. Bibcode:2023arXiv231116629S
    .
17. ^
    Zbl 0633.52006
    .
18. ^ Webb, Robert. "Enumeration of Stellations". Stella. Archived from the original on 2022-11-26. Retrieved 2023-09-21.
19. S2CID 202679388.{{cite book}}: CS1 maint: location missing publisher (link
    )
20. Zbl 1476.51020
    .
21. 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. Zbl 1004.20003
    .