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De Polignac–Legendre formula

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The prime factorization of
n!, n   ≥   1,
is given by de Polignac's formula, named after Alphonse de Polignac. L. E. Dickson attributes the formula to Legendre.[1]

Formula

The prime factorization of 
n!
, 
n   ≥   1
, is given by

where

is the p-adic valuation of
n!
(order of prime
p
in the prime factorization of
n!
) and the brackets represent the floor function.

Examples

Consider

for which the
p
-adic valuations for primes
p
up to 5 are

Notes

  1. Leonard Eugene Dickson, History of the Theory of Numbers, Volume 1, Carnegie Institution of Washington, 1919, page 263.