Definitions

# De Polignac's formula

In number theory, de Polignac's formula, named after Alphonse de Polignac, gives the prime decomposition of the factorial n!, where n ≥ 1 is an integer.

## The formula

Let n ≥ 1 be an integer. Then the prime decomposition of n! is given by

$prod_\left\{p, mathrm\left\{prime\right\},, p leq n\right\}\left\{p^\left\{s_p\left(n\right)\right\}\right\}$

where

$s_p\left(n\right) = sum_\left\{j = 1\right\}^inftyleftlfloorfrac\left\{n\right\}\left\{p^j\right\}rightrfloor$

and the brackets represent the floor function.

Note that, for any real number x, and any integer n, we have:

$leftlfloorfrac\left\{x\right\}\left\{n\right\}rightrfloor = leftlfloorfrac\left\{lfloor\left\{x\right\}rfloor\right\}\left\{n\right\}rightrfloor$

which allows one to more easily compute the terms sp(n).

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