Legendre's formula

For a prime number

p

and a positive integer

n

, the P-adic valuation of a factorial is

v_{p} (n!) = \sum_{j \geq 1} ⌊ \frac{n}{p ^{j}} ⌋ .

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Legendre's formula by

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Legendre's formula, also known as Legendre's theorems or Legendre's formula for finding the exponent of a prime \( p \) in the factorization of \( n! \) (n factorial), provides a way to determine how many times a prime number divides \( n! \).

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