63. Products of primitive roots modulo p.py

def products_of_primitive_roots_mod_p(p):
  """
  Finds the products of all pairs of primitive roots modulo p.

  Args:
    p: The modulus.

  Returns:
    A list of the products of all pairs of primitive roots modulo p.
  """

  # Get the list of primitive roots modulo p.
  primitive_roots_mod_p = primitive_roots_mod_p(p)

  # Initialize a list to store the products of all pairs of primitive roots modulo p.
  products_of_primitive_roots_mod_p = []

  # Iterate over all pairs of primitive roots modulo p.
  for i in range(len(primitive_roots_mod_p)):
    for j in range(i + 1, len(primitive_roots_mod_p)):
      products_of_primitive_roots_mod_p.append(primitive_roots_mod_p[i] * primitive_roots_mod_p[j])

  return products_of_primitive_roots_mod_p

def primitive_roots_mod_p(p):
  """
  Finds all primitive roots modulo p.

  Args:
    p: The modulus.

  Returns:
    A list of primitive roots modulo p.
  """

  # Create a list to store the primitive roots modulo p.
  primitive_roots_mod_p = []

  # Iterate over all integers from 2 to p - 1.
  for i in range(2, p):

    # If the integer is a primitive root modulo p, add it to the list.
    if is_primitive_root(i, p):
      primitive_roots_mod_p.append(i)

  return primitive_roots_mod_p

def is_primitive_root(a, p):
  """
  Checks if the integer a is a primitive root modulo p.

  Args:
    a: The integer.
    p: The modulus.

  Returns:
    True if the integer a is a primitive root modulo p, False otherwise.
  """

  # Check if a is an integer greater than 1 and less than p.
  if a <= 1 or a >= p:
    return False

  # Check if the order of a modulo p is p - 1.
  order_of_a = order_of_integer(a, p)
  if order_of_a != p - 1:
    return False

  # Check if a is a primitive root modulo p.
  for i in range(2, p):
    if pow(a, i) % p == 1:
      return False

  return True