Write Sage Script to Calculate Number of Primitive Roots

Primitive roots are essential in number theory and cryptography. This guide explains how to write a SageMath script to calculate the number of primitive roots modulo a prime number, including the formula, example, and calculator.

What Are Primitive Roots?

A primitive root modulo n is an integer g that generates all the integers modulo n through repeated exponentiation. In other words, g is a primitive root modulo n if the smallest positive integer k such that g^k ≡ 1 mod n is k = φ(n), where φ is Euler's totient function.

Primitive roots exist only for certain moduli. Specifically, for a prime p, there are exactly φ(p-1) primitive roots modulo p. For a prime power p^k, the number of primitive roots is φ(p^(k-1)(p-1)).

Sage Script to Calculate Primitive Roots

Below is a SageMath script that calculates the number of primitive roots modulo a given prime number:

def number_of_primitive_roots(p):
    """
    Calculate the number of primitive roots modulo a prime p.

    Input:
        p (integer): A prime number

    Output:
        Integer representing the number of primitive roots modulo p
    """
    if not p.is_prime():
        raise ValueError("Input must be a prime number")
    return euler_phi(p - 1)

The script uses Euler's totient function φ(n) to determine the number of primitive roots. For a prime p, φ(p-1) gives the exact count.

Example Calculation

Let's calculate the number of primitive roots modulo 7:

Step 1: Verify that 7 is prime (it is).

Step 2: Calculate φ(7-1) = φ(6).

Step 3: φ(6) = 2 (since 6 has two numbers less than it that are coprime: 1 and 5).

Therefore, there are 2 primitive roots modulo 7.

The primitive roots modulo 7 are 3 and 5.

Formula Used

The number of primitive roots modulo a prime p is given by Euler's totient function φ(p-1):

Number of primitive roots modulo p = φ(p-1)

Where φ(n) is the number of integers up to n that are coprime with n.

FAQ

What is the difference between primitive roots and generators?: Primitive roots and generators are essentially the same concept. Both refer to elements that generate the multiplicative group of integers modulo n.
Can I find primitive roots modulo a composite number?: Primitive roots exist only for certain moduli. For composite numbers, they exist only if the modulus is 2, 4, p^k, or 2p^k, where p is an odd prime.
How do I verify if a number is a primitive root?: To verify if g is a primitive root modulo n, check that the smallest positive integer k such that g^k ≡ 1 mod n is k = φ(n).