In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, de Polignac's formula, named after Alphonse de Polignac

Alphonse de Polignac

Alphonse de Polignac was a French mathematician. In 1849 he made Polignac's conjecture:The case k = 1 is the twin prime conjecture.-References:...

, gives the prime decomposition of the factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 n!, where n ≥ 1 is an integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

. L. E. Dickson

Leonard Eugene Dickson

Leonard Eugene Dickson was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory.-Life:Dickson considered himself a Texan by...

 attributes the formula to Legendre

Adrien-Marie Legendre

Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

.

The formula

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


where


and the brackets represent the floor function

Floor function

In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...

.

Note that, for any real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

x, and any integer n, we have:


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