AKS Primes Algorithm in Python: A Comprehensive Guide
Prime numbers have always been a fascinating topic in the world of mathematics. They are the building blocks of all numbers and have a unique set of properties that make them stand out. The search for prime numbers has been an ongoing pursuit for centuries, and with the advent of computers, new and efficient algorithms have been developed to identify and generate these elusive numbers. One such algorithm is the AKS Primes Algorithm, which has gained popularity in recent years for its simplicity and efficiency. In this article, we will explore the inner workings of this algorithm and how it can be implemented in Python.
What is the AKS Primes Algorithm?
The AKS Primes Algorithm is a deterministic algorithm for testing the primality of a given number. It was developed in 2002 by Indian mathematicians Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, and the acronym "AKS" is derived from their initials. This algorithm falls under the category of polynomial-time algorithms, meaning it can determine if a number is prime in polynomial time, making it significantly faster than traditional methods.
The main idea behind the AKS Primes Algorithm is to use Fermat's Little Theorem and Euler's Totient Function to prove the primality of a number. The algorithm takes advantage of the fact that for any prime number p, the following equation always holds true: (x + a)^p ≡ x^p + a (mod p), where a is any integer and x is a variable. This equation is known as the binomial expansion of (x + a)^p. The algorithm also uses the fact that for any composite number n, there exists an integer a such that the above equation does not hold true. This is the basis of the AKS Primes Algorithm, and it is what makes it so efficient.
Implementing AKS Primes Algorithm in Python
Now, let's see how we can implement the AKS Primes Algorithm in Python. The first step is to import the necessary libraries for our code, which in this case, are the math and sympy libraries.
```
import math
from sympy import prime, isprime
```
Next, we define a function called `is_prime` that takes in a number as an argument and returns `True` or `False` based on whether the number is prime or not. We will use this function to check if the algorithm correctly identifies prime and composite numbers.
```
def is_prime(n):
# Check if n is prime using sympy library
if isprime(n):
return True
else:
return False
```
Now, we can define the AKS Primes Algorithm function, which takes in a number `n` as an argument and returns `True` or `False` based on whether the number is prime or not.
```
def AKS_primes(n):
# Check if n is a perfect power
for i in range(2, int(math.log2(n)) + 1):
root = n ** (1 / i)
# Check if the root is an integer
if root.is_integer():
# If root is an integer, n is not prime
