63 (number)

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63 is a composite number and a Woodall number^[1] and a Central Delannoy numbers.^[2]

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 ${\displaystyle a^{6}-b^{6}=2^{6}-1^{6}=63=3^{2}\times 7=(a^{2}-b^{2})^{2}(a^{3}-b^{3})}$.^[3]

In the integer positive definite quadratic matrix ${\displaystyle \{1,2,3,5,6,7,10,14,15\}}$ representative of all (even and odd) integers,^[4][5] the sum of all nine terms is equal to 63.

Finite simple groups

In the classification of finite simple groups of Lie type, 63 is an exponents that figures in the orders of three exceptional groups of Lie type. Lie algebra ${\displaystyle E_{7}}$ holds sixty-three positive root vectors in the seven-dimensional space.^[6]

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),^[7] 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.^[7] 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.^[8][9][10]