Legendre's Formula

Revision as of 07:27, 9 November 2015 by Chocopuff (talk | contribs) (Part 2)

Legendre's Formula states that

\[e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}\]

where $p$ is a prime and $e_p(n)$ is the exponent of $p$ in the prime factorization of $n$ and $S_p(n)$ is the sum of the digits of $n$ when written in base $p$.


Part 1

We use a counting argument.

We could say that $e_p(n!)$ is equal to the number of multiples of $p$ less than $n$, or $\lfloor \frac{n}{p}\rfloor$. But the multiples of $p^2$ are only counted once, when they should be counted twice. So we need to add $\lfloor \frac{n}{p^2}\rfloor$ on. But this only counts the multiples of $p^3$ twice, when we need to count them thrice. Therefore we must add a $\lfloor \frac{n}{p^3}\rfloor$ on. We continue like this to get $e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor$. This makes sense, because the terms of this series tend to 0.

Part 2

Let the base $p$ representation of $n$ be \[e_xe_{x-1}e_{x-2}\dots e_0\] where the $e_i$ are digits in base $p.$ Then, the base $p$ representation of $\lfloor \frac{n}{p^i}\rfloor$ is \[e_xe_{x-1}\dots e_{x-i}.\] Note that the infinite sum of these numbers (which is $e_p(n!)$) 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}

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