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[parent] factorial modulo prime powers (Result)

For $ n \in \mathbb{N}$ and any prime number $ p$, $ \left(n!\right)_p$ is the product of numbers $ 1 \le m \le n$, where $ p \not\vert m$.

For natural numbers $ n, s$ and a given prime number $ p$, we have the congruence

$\displaystyle \frac{n!}{p^{\sum\limits_{i=0}^d \left\lfloor \frac{n}{p^i}\right... ...eft\lfloor\frac{n}{p^s+i}\right\rfloor} \left(N_i!\right)_p\right) \pmod{p^s}, $
where $ N_i$ is the least non-negative residue of $ \left\lfloor \frac{n}{p^i}\right\rfloor \pmod{p^s}$. Here $ d+1$ denotes the number of digits in the representation of $ n$ in base $ p$. More precisely, $ \pm 1$ is $ -1$ unless $ p=2, s \ge 3$.
Proof. Let $ i \ge 0$. Then the set of numbers between 1 and $ \left\lfloor \frac{n}{p^i}\right\rfloor$ is
$\displaystyle \left\{kp, k \ge 1, k \le \left\lfloor\frac{n}{p^{i+1}}\right\rfloor\right\}.$
This is true for every integer $ i$ with $ p^{i+1} \le n$. So we have
$\displaystyle \frac{\left\lfloor \frac{n}{p^i}\right\rfloor!}{\left\lfloor \fra... ...^{i+1}}\right\rfloor}} =\left(\left\lfloor\frac{n}{p^i}\right\rfloor!\right)_p.$ (1)

Multiplying all terms with $ 0 \le i \le d$, where $ d$ is the largest power of $ p$ not greater than $ n$, the statement follows from the generalization of Anton's congruence. $ \qedsymbol$



"factorial modulo prime powers" is owned by Thomas Heye.
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Cross-references: Anton's congruence, terms, integer, base, representation, digits, residue, congruence, natural numbers, numbers, product, prime number

This is version 4 of factorial modulo prime powers, born on 2003-01-23, modified 2004-04-01.
Object id is 3915, canonical name is FactorialModulePrimePowers.
Accessed 2427 times total.

Classification:
AMS MSC: 11A07 (Number theory :: Elementary number theory :: Congruences; primitive roots; residue systems)
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