Difference between revisions of "Legendre's Formula"
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where <math>p</math> is a prime and <math>e_p(n)</math> is the [[exponent]] of <math>p</math> in the [[prime factorization]] of <math>n</math> and <math>S_p(n)</math> is the [[sum]] of the [[digit]]s of <math>n</math> when written in [[base]] <math>p</math>. | where <math>p</math> is a prime and <math>e_p(n)</math> is the [[exponent]] of <math>p</math> in the [[prime factorization]] of <math>n</math> and <math>S_p(n)</math> is the [[sum]] of the [[digit]]s of <math>n</math> when written in [[base]] <math>p</math>. | ||
− | == | + | ==Proofs== |
=== Part 1 === | === Part 1 === | ||
+ | We use a counting argument. | ||
+ | |||
We could say that <math>e_p(n!)</math> is equal to the number of multiples of <math>p</math> less than <math>n</math>, or <math>\lfloor \frac{n}{p}\rfloor</math>. But the multiples of <math>p^2</math> are only counted once, when they should be counted twice. So we need to add <math>\lfloor \frac{n}{p^2}\rfloor</math> on. But this only counts the multiples of <math>p^3</math> twice, when we need to count them thrice. Therefore we must add a <math>\lfloor \frac{n}{p^3}\rfloor</math> on. We continue like this to get <math>e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor</math>. This makes sense, because the terms of this series tend to 0. | We could say that <math>e_p(n!)</math> is equal to the number of multiples of <math>p</math> less than <math>n</math>, or <math>\lfloor \frac{n}{p}\rfloor</math>. But the multiples of <math>p^2</math> are only counted once, when they should be counted twice. So we need to add <math>\lfloor \frac{n}{p^2}\rfloor</math> on. But this only counts the multiples of <math>p^3</math> twice, when we need to count them thrice. Therefore we must add a <math>\lfloor \frac{n}{p^3}\rfloor</math> on. We continue like this to get <math>e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor</math>. This makes sense, because the terms of this series tend to 0. | ||
=== Part 2 === | === Part 2 === | ||
− | Let the base <math>p</math> representation of <math>n</math> be < | + | Let the base <math>p</math> representation of <math>n</math> be <cmath>e_xe_{x-1}e_{x-2}\dots e_0</cmath> where the <math>e_i</math> are digits in base <math>p.</math> Then, the base <math>p</math> representation of <math>\lfloor \frac{n}{p^i}\rfloor</math> is <cmath>e_xe_{x-1}\dots e_{x-i}.</cmath> Note that the infinite sum of these numbers (which is <math>e_p(n!)</math>) is |
− | |||
− | |||
− | < | + | <cmath>\begin{aligned} \sum_{j=1}^{x} e_j(p^{j-1}+p^{j-2}+\cdots +1) &= \sum_{j=1}^{x} e_j(\frac{p^j-1}{p-1}) \\ &=\frac{\sum_{j=1}^{x} e_jp^j -\sum_{j=1}^{x} e_j}{p-1} \\ &=\frac{(n-e_0)-(S_p(n)-e_0)}{p-1}=\frac{n-S_p(n)}{p-1}. \end{aligned}</cmath> |
{{stub}} | {{stub}} |
Revision as of 11:07, 25 November 2013
Legendre's Formula states that
where is a prime and
is the exponent of
in the prime factorization of
and
is the sum of the digits of
when written in base
.
Proofs
Part 1
We use a counting argument.
We could say that is equal to the number of multiples of
less than
, or
. But the multiples of
are only counted once, when they should be counted twice. So we need to add
on. But this only counts the multiples of
twice, when we need to count them thrice. Therefore we must add a
on. We continue like this to get
. This makes sense, because the terms of this series tend to 0.
Part 2
Let the base representation of
be
where the
are digits in base
Then, the base
representation of
is
Note that the infinite sum of these numbers (which is
) is
\begin{aligned} \sum_{j=1}^{x} e_j(p^{j-1}+p^{j-2}+\cdots +1) &= \sum_{j=1}^{x} e_j(\frac{p^j-1}{p-1}) \\ &=\frac{\sum_{j=1}^{x} e_jp^j -\sum_{j=1}^{x} e_j}{p-1} \\ &=\frac{(n-e_0)-(S_p(n)-e_0)}{p-1}=\frac{n-S_p(n)}{p-1}. \end{aligned} (Error compiling LaTeX. Unknown error_msg)
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