Step by Step solution about How to resolve the algorithm Totient function step by step in the Python programming language

The code is written in Python, and it calculates the Euler's totient function (φ) for a given positive integer n. Euler's phi function counts the number of positive integers less than or equal to n that are coprime to n, which means that they have no common divisors other than 1. The code defines a function called φ that takes a positive integer n as an argument and returns the number of positive integers less than or equal to n that are coprime to n. The function uses a generator expression to iterate over the numbers from 1 to n, and for each number k, it checks if k is coprime to n by using the gcd function from the math module. If k is coprime to n, then the generator expression yields 1. The sum function is then used to sum the values yielded by the generator expression, which gives the number of positive integers less than or equal to n that are coprime to n.

The main part of the code tests the φ function for different values of n. It first defines a function called is_prime that checks if a given positive integer n is prime. A prime number is a positive integer that is only divisible by 1 and itself. The is_prime function checks if φ(n) is equal to n - 1. If it is, then n is prime, otherwise it is not.

The main part of the code then iterates over the numbers from 1 to 25, and for each number n, it prints the value of φ(n) and whether or not n is prime.

The main part of the code also calculates the number of prime numbers up to certain limits. It iterates over the numbers from 1 to 10,000, and for each number n, it checks if n is prime by using the is_prime function. If it is, then the count of prime numbers is incremented. The count of prime numbers is then printed for the limits 100, 1000, and 10,000.

from math import gcd

def  φ(n):
    return sum(1 for k in range(1, n + 1) if gcd(n, k) == 1)

if __name__ == '__main__':
    def is_prime(n):
        return φ(n) == n - 1
    
    for n in range(1, 26):
        print(f" φ({n}) == {φ(n)}{', is prime' if is_prime(n)  else ''}")
    count = 0
    for n in range(1, 10_000 + 1):
        count += is_prime(n)
        if n in {100, 1000, 10_000}:
            print(f"Primes up to {n}: {count}")