A formula giving the exponent of a prime p dividing n!: v_p(n!) = sum_{k≥1} floor(n/p^k), i.e., the p-adic valuation of n! equals the sum of the integer parts of n divided by successive powers of p until those terms vanish.

Principle

Count contributions of multiples of p, p^2, p^3, … to n!: each multiple of p contributes at least one factor p, multiples of p^2 contribute an extra factor, and so on; the floor-sum aggregates these contributions exactly.

Demonstration

Example: for n=10 and p=2, v_2(10!) = floor(10/2)+floor(10/4)+floor(10/8)+floor(10/16)+… =5+2+1+0+…=8, so 2^8 divides 10! but 2^9 does not.

Misapplication

Applying the infinite sum without taking floors or forgetting that terms eventually vanish; or trying to use the formula for objects other than factorials (e.g., arbitrary binomial products) without verifying combinatorial structure.

Consequence

Provides an efficient and exact method to compute p-adic valuations of factorials, which is central in divisibility arguments, evaluating binomial coefficient valuations, and in p-adic and combinatorial number theory.

Reversal

Instead of summing floor(n/p^k), one can view valuations combinatorially via carry count in base p addition (Kummer's theorem), offering a dual perspective: Legendre counts multiples, Kummer counts carries in addition of binomial indices.

Boundary

The formula applies to n! for nonnegative integers n and primes p; it does not directly generalize to non-integer factorial analogues or to valuation of arbitrary integer sequences without appropriate combinatorial decomposition.

Semantic Tension

Close relation and occasional confusion with Kummer's theorem and with p-adic valuation techniques: Legendre gives a direct floor-sum, while Kummer characterizes valuations of binomial coefficients by base-p carries, two complementary but distinct viewpoints.

Synthesis

Legendre's Formula exactly expresses the p-adic valuation of n! as the sum of floors of n divided by powers of p, encoding multiplicity counts of prime powers in factorials and serving as a practical computational tool in divisibility and p-adic investigations.