Legendre's formula

Legendre's formula gives the exponent of a prime in the prime factorization of a factorial by summing the integer parts of successive quotients of the factorial's argument by powers of that prime.

Prime factorization: The expression of an integer as a product of prime powers. The formula computes one prime's exponent in this decomposition.
Binomial coefficient: The number of ways to choose k objects from n, given by n! divided by k!(n−k)!. Subtracting three factorial valuations gives each prime's exponent in a binomial coefficient.
Adrien-Marie Legendre: A French mathematician whose work included number theory, analysis, and the method of least squares. The factorial valuation formula is named after him.
Prime number: An integer greater than one with no positive divisors other than one and itself. The formula takes a prime as its input and counts its power in a factorial.
Valuation (mathematics): A function that measures divisibility, often by recording the exponent of a prime in an integer. Legendre's formula calculates the prime-adic valuation of a factorial.
Trailing zeros of a factorial: The number of terminal zeroes in a factorial's decimal representation. The decimal zero count is the smaller of the factorial's exponents of 2 and 5.
Théorie des nombres: Legendre's 1830 treatise on number theory, which presented results including the factorial valuation formula. The formula appeared in this work.
Integer part: The greatest integer less than or equal to a real number, also called its floor. Taking integer parts counts multiples without including incomplete groups.
Floor function: The function assigning each real number the greatest integer less than or equal to it. Each summand is the floor of the factorial argument divided by a prime power.
Central binomial coefficient: The coefficient C(2n,n), which is the largest entry in the nth row of Pascal's triangle. Legendre's formula determines its prime divisibility and supports bounds on its size.

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