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Difference between revisions of "Talk:Twenty-four"

(Created page with "I would like to edit this page to add some additional interesting information about the number <math>24</math>. <math>24</math> is, in fact, the difference of squares in two...")
 
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Also, the fact that <math>24 = 4!</math> is noteworthy, since for example <math>4!</math> is the order of <math>S_4</math>, the group of permutations of four objects or of orientation-preserving symmetries of a cube or an octahedron.
 
Also, the fact that <math>24 = 4!</math> is noteworthy, since for example <math>4!</math> is the order of <math>S_4</math>, the group of permutations of four objects or of orientation-preserving symmetries of a cube or an octahedron.
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Another interesting, if advanced, piece of information is that the definition of Ramanujan's tau function includes a conspicuous power of <math>24</math>. <math>\tau(n)</math> is the coefficient of the degree-<math>n</math> term of the power series <cmath>q \left( (1-q)(1-q^2)(1-q^3) \dots \right)^{24}.</cmath> Notably, <math>\tau</math> is multiplicative, that is, if <math>m</math> and <math>n</math> are relatively prime, then <math>\tau(m)\tau(n) = \tau(mn)</math>. [[User:Orange quail 9|Orange quail 9]] ([[User talk:Orange quail 9|talk]]) 12:34, 18 May 2022 (EDT)

Latest revision as of 11:34, 9 December 2022

I would like to edit this page to add some additional interesting information about the number $24$.

$24$ is, in fact, the difference of squares in two ways: $24 = 7^2 - 5^2 = 5^2 - 1^2$. In fact, it is the common difference of the smallest nontrivial arithmetic progression among the perfect squares: $1 \rightarrow 25 \rightarrow 49$. $24$ is not the sum of any two squares, however.

Also, the fact that $24 = 4!$ is noteworthy, since for example $4!$ is the order of $S_4$, the group of permutations of four objects or of orientation-preserving symmetries of a cube or an octahedron.

Another interesting, if advanced, piece of information is that the definition of Ramanujan's tau function includes a conspicuous power of $24$. $\tau(n)$ is the coefficient of the degree-$n$ term of the power series \[q \left( (1-q)(1-q^2)(1-q^3) \dots \right)^{24}.\] Notably, $\tau$ is multiplicative, that is, if $m$ and $n$ are relatively prime, then $\tau(m)\tau(n) = \tau(mn)$. Orange quail 9 (talk) 12:34, 18 May 2022 (EDT)

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