Lagmental Vicfred

Legendre's Formula Counts p in a Factorial by Vicfred

tags: mathematics, p-adic numbers, valuations, Hensel lemma

The p-adic valuation of n factorial is the sum of all quotient floors by powers of p. A small computation will anchor the general statement before the abstraction takes over.

Objects and notation

The valuation \(v_p(n)\) counts factors of \(p\), and the metric \(|x|_p=p^{-v_p(x)}\) reverses the usual sense of size. Hensel lifting turns approximate roots modulo \(p\) into exact \(p\)-adic roots.

$$ v_p(n!)=\sum_{k\ge1}\left\lfloor\frac n{p^k}\right\rfloor $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ v_p(n!)=\frac{n-s_p(n)}{p-1} $$

Push the symbols

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ v_5(100!)=\left\lfloor\frac{100}{5}\right\rfloor+\left\lfloor\frac{100}{25}\right\rfloor=20+4=24 $$

Structural reading

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad v_p(n!)=\sum_{k\ge1}\left\lfloor\frac n{p^k}\right\rfloor,\\[5pt] \mathsf{C}\;&:\quad v_p(n!)=\frac{n-s_p(n)}{p-1}. \end{aligned} $$

A hypothesis worth keeping

A congruence root lifts uniquely only in the simple-root case. Multiple roots need stronger inequalities and may split, disappear, or lift nonuniquely.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] v_p(n!)=\frac{n-s_p(n)}{p-1} \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Thu 03 August 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.