De Polignac's formula

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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.

[edit] The formula

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

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

where

s_p(n) = \sum_{j = 1}^\infty\left\lfloor\frac{n}{p^j}\right\rfloor

and the brackets represent the floor function.

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

\left\lfloor\frac{x}{n}\right\rfloor = \left\lfloor\frac{\lfloor{x}\rfloor}{n}\right\rfloor

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

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