Difference between revisions of "Legendre's Formula"
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'''Legendre's formula''' states that | '''Legendre's formula''' states that | ||
| − | <cmath>e_p(n)=\sum_{i\geq 1} \lfloor \dfrac{n}{p^ | + | <cmath>e_p(n)=\sum_{i\geq 1} \lfloor \dfrac{n}{p^i}\rfloor =\frac{n-S_{p}(n)}{p-1}</cmath> |
where <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 digits of n when written in base <math>p</math>. | where <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 digits of n when written in base <math>p</math>. | ||
Revision as of 10:17, 5 August 2008
Legendre's formula states that
![\[e_p(n)=\sum_{i\geq 1} \lfloor \dfrac{n}{p^i}\rfloor =\frac{n-S_{p}(n)}{p-1}\]](http://latex.artofproblemsolving.com/f/0/3/f03dec742f40ad2ec5ece3869ca67bb53514fbe4.png)
where 
is the exponent of 
in the prime factorization of 
, and 
is the sum of the digits of n when written in base .
Proof
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