|
|
| Line 1: |
Line 1: |
| In [[number theory]], '''de Polignac's formula''', named after [[Alphonse de Polignac]], gives the [[prime decomposition]] of the [[factorial]] ''n''<nowiki>!</nowiki>, where ''n'' ≥ 1 is an [[integer]]. [[Leonard Eugene Dickson|L. E. Dickson]] attributes the formula to [[Adrien-Marie Legendre|Legendre]].<ref>[[Leonard Eugene Dickson]], ''[[History of the Theory of Numbers]]'', Volume 1, Carnegie Institution of Washington, 1919, page 263.</ref>
| | Hello. Let me [http://Www.slideshare.net/realtrafficsource/celebrities-with-herpes introduce] the author. Her title is Emilia Shroyer but it's not the most female title out there. Minnesota has usually been his home but std testing at home his wife wants them to transfer. over the counter std test To gather coins is a thing that I'm completely addicted to. Hiring has been my occupation for some time but I've already applied for another 1.<br><br>Have a look at my weblog at home [http://www.womentowomen.com/sex-fertility/enjoying-safe-sex-in-midlife/ std testing] ([http://ddra1.com/xe/board_JEIV43/275161 you can try this out]) |
| | |
| ==The formula==
| |
| Let ''n'' ≥ 1 be an integer. The prime decomposition of ''n''! is given by
| |
| | |
| :<math>n! = \prod_{\text{prime }p\le n} p^{s_p(n)}, </math>
| |
| | |
| where
| |
| | |
| :<math>s_p(n) = \sum_{j = 1}^\infty \left\lfloor\frac{n}{p^j}\right\rfloor, </math>
| |
| | |
| and the brackets represent the [[floor function]]. Note that the former product can equally well be taken only over primes less than or equal to ''n'', and the latter sum can equally well be taken for ''j'' ranging from ''1'' to log<sub>''p''</sub>(''n''), i.e :
| |
| | |
| :<math>s_p(n) = \sum_{j = 1}^{\lfloor \log_p(n) \rfloor} \left\lfloor\frac{n}{p^j}\right\rfloor </math>
| |
| | |
| Note that, for any [[real number]] ''x'', and any integer ''n'', we have:
| |
| | |
| :<math>\left\lfloor\frac{x}{n}\right\rfloor = \left\lfloor\frac{\lfloor x \rfloor}{n}\right\rfloor</math>
| |
| | |
| which allows one to more easily compute the terms ''s''<sub>''p''</sub>(''n'').
| |
| | |
| The small disadvantage of the De Polignac's formula is that '''we need to know all the primes up to ''n'''''.
| |
| In fact,
| |
| :<math>\displaystyle n! = \prod_{i=1}^{\pi(n)} p_{i}^{s_{p_{i}}(n)} = \prod_{i=1}^{\pi(n)} p_i^{ \sum_{j = 1}^{\lfloor \log_{p_i}(n) \rfloor} \left\lfloor\frac{n}{{p_i}^j}\right\rfloor } </math>
| |
| | |
| where <math>\pi(n)</math> is a [[prime-counting function]] counting the number of prime numbers less than or equal to ''n''
| |
| | |
| == Notes and references ==
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:De Polignac's Formula}}
| |
| [[Category:Number theory]] | |
| [[Category:Factorial and binomial topics]]
| |
Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most female title out there. Minnesota has usually been his home but std testing at home his wife wants them to transfer. over the counter std test To gather coins is a thing that I'm completely addicted to. Hiring has been my occupation for some time but I've already applied for another 1.
Have a look at my weblog at home std testing (you can try this out)