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单词 ThePrimePowerDividingAFactorial
释义

the prime power dividing a factorial


In 1808, Legendre showed that the exact power of a prime p dividingn! is

i=1Knpi

where K is the largest power of p being n.

Proof.

If p>n then p doesn’t divide n!, and its power is 0, and the sumabove isempty. So let the prime pn.
For each 1iK, there are npi-npi+1 numbers between 1 andn withi being the greatest power of p dividing each. So the power of pdividing n! is

i=1Ki(npi-npi+1).

But each npi,i2 in thesum appears with factors iand i-1, so the above sum equals

i=1Knpi.

Corollary.
k=1KnpK=n-δp(n)p-1,

where δp denotes the sums of digits function in base p.

Proof.

If n<p, then δp(n)=n and npis 0=n-δp(n)p-1. So we assume pn.

Let nKnK-1n0 be the p-adic representation of n. Then

n-δp(n)p-1=k=1Knk(j=0k-1pj)
=0j<kKpj.nk
=(n1+n2p++nKpK-1)
+(n2+n3p++nKpK-2)
+nK
=k=1Knpk.

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更新时间:2025/5/4 17:31:25