Understanding the Highest Power of 2 Dividing n! for n 2^k

In this article, we will explore the concept of the highest power of 2 that can divide a factorial, specifically when n is a power of 2. This is an essential topic in number theory and relevant for those studying combinatorics, discrete mathematics, and algorithm design. We will delve into the mathematics behind it and provide clear examples and explanations.

The Problem Statement

Given a number m 4096 2^12, we need to find the largest power of 2 that divides m! 4096! 2^12!. This involves understanding how the powers of 2 contribute to the factorization of factorials.

Using the Divisor Function to Find the Exponent of 2

The exponent of the largest power of 2 that divides n! can be found using the sum:

m/2 m/2^2 m/2^3 ….until the power of 2 exceeds m.

In our case, we are dealing with m 2^12 4096. This sum computes to:

2048 1024 512 256 128 64 32 16 8 4 2 1 4095 2^12 - 1

This result is derived from the formula for the sum of a finite geometric progression, as each term in the sum is decreasing by powers of 2.

Largest Power of 2 in Factorials

We observe that the powers of 2 less than or equal to 2^12 are:

2^1 1024 2^11 2048 2^12 4096 (our given m) 2^13 8192 (exceeds m)

The function n_2(n!) (or f_n) represents the largest power of 2 that divides n!. The formula n_2(2^k!) is given by:

2^k/2 2^k/2^2 2^k/2^3 … 2^k/2^k-1 2^k - 1

Hence, for n 1024 (2^1), we have:

n_2(1024!) 2^1 - 1 1023

Similarly, for higher values:

n 2048 (2^11): n_2(2048!) 2^11 - 1 2047 n 4096 (2^12): n_2(4096!) 2^12 - 1 4095 n 8192 (2^13): n_2(8192!) 2^13 - 1 8191

Generalizing for Any Natural Number n

For a general natural number n, we can see that:

n_2(n!) n_2(n1!) for n1

This implies:

n_2(1000!) 994 * (2^1023 - 1) 994 * (2^11 - 1)

Continuing the pattern, we get:

n_2(21!) 2^1023 - 1

n_2(211!) 2^1023 - 1 * (2^1022 - 1)

And so on, leading to:

n_2(212!) 2^1023 - 1 * (2^1022 - 1) * (2^1021 - 1) … * (2^1 - 1) 2^1023 - 1

And for the highest factor:

n_2(9999!) 2^9999 - 1 (where stands for 'far less than')

Conclusion

The problem initially posed was asking for the highest power of 2 dividing 4096!, which we demonstrated to be 2^12 - 1. However, the real lesson is how to find the highest power of 2 dividing any factorial n!. The method involves summing the decreasing terms of a geometric progression and understanding the properties of factorials.

This article has covered the key concepts and provided a detailed breakdown of the mathematical steps. For further reading and deeper understanding, consult textbooks on number theory and discrete mathematics.

Key Takeaways:

The highest power of 2 dividing n! is given by n_2(n!). The formula for the sum of powers of 2 to find n_2(n!) is 2^k - 1, where k is the highest power of 2 in n! The process involves summing a geometric series and understanding the factorization of factorials.