Exponent of Prime ( p ) in ( n! )

When we talk about the exponent of a prime ( p ) in ( n! ) (n factorial), we are referring to the number of times the prime number ( p ) divides into ( n! ). This is an important concept in number theory and combinatorics, as it helps us understand the prime factorization of factorial numbers.

Understanding Factorials

Before diving into the exponent of a prime in a factorial, let's understand what a factorial is. 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! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 ]

For example:

[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 ]

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of ( 120 ) is ( 2^3 \times 3 \times 5 ).

Exponent of Prime ( p ) in ( n! )

To find the exponent of a prime ( p ) in ( n! ), we use the following formula, often attributed to the mathematician Christian Kramp:

[ e_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor ]

Where ( e_p(n!) ) is the exponent of ( p ) in ( n! ), and ( \left\lfloor x \right\rfloor ) denotes the floor function, which gives the greatest integer less than or equal to ( x ).

Why This Formula Works

The formula works because each multiple of ( p ) contributes at least one factor of ( p ) to ( n! ). Multiples of ( p^2 ) contribute an additional factor, multiples of ( p^3 ) contribute yet another, and so on. By summing the floor of ( n ) divided by each power of ( p ), we count each contribution of ( p ) to the factorial.

Table of Differences and Important Points

Point	Description
Factorial	The product of all positive integers up to ( n ).
Prime Factorization	Expressing a number as the product of its prime factors.
Exponent of ( p )	The number of times ( p ) appears in the prime factorization.
Kramp's Formula	A method to calculate the exponent of ( p ) in ( n! ).

Examples

Let's go through some examples to illustrate how to use the formula.

Example 1: Exponent of 2 in 10!

Calculate the exponent of the prime number 2 in ( 10! ).

[ e_2(10!) = \left\lfloor \frac{10}{2} \right\rfloor + \left\lfloor \frac{10}{2^2} \right\rfloor + \left\lfloor \frac{10}{2^3} \right\rfloor + \ldots ]

[ e_2(10!) = \left\lfloor \frac{10}{2} \right\rfloor + \left\lfloor \frac{10}{4} \right\rfloor + \left\lfloor \frac{10}{8} \right\rfloor ]

[ e_2(10!) = 5 + 2 + 1 = 8 ]

So, the exponent of 2 in ( 10! ) is 8.

Example 2: Exponent of 5 in 100!

Calculate the exponent of the prime number 5 in ( 100! ).

[ e_5(100!) = \left\lfloor \frac{100}{5} \right\rfloor + \left\lfloor \frac{100}{5^2} \right\rfloor + \left\lfloor \frac{100}{5^3} \right\rfloor + \ldots ]

[ e_5(100!) = \left\lfloor \frac{100}{5} \right\rfloor + \left\lfloor \frac{100}{25} \right\rfloor ]

[ e_5(100!) = 20 + 4 = 24 ]

So, the exponent of 5 in ( 100! ) is 24.

Conclusion

Understanding the exponent of a prime ( p ) in ( n! ) is crucial for solving problems in number theory and combinatorics. The formula provided by Kramp offers a systematic way to calculate this exponent, which is essential for determining the divisibility properties of factorial numbers. By practicing with examples, one can become proficient in applying this concept to various mathematical problems.