Linear congruential generator testing full period

Question

Write a program to demonstrate that for a linear congruential generator with modulus = 2 and constant = 1, in order to achieve the full period, the multiplier must be equal to 4 + 1. Consider =5, 7, and 10; and =2 and 9. You should also consider two values of the multiplier that do not match this.

My problem with this is that I have the code to run this in Python. The suggestions I have been given are

def lcg(modulus, a, c, seed):
    """Linear congruential generator."""
    while True:
        seed = (a * seed + c) % modulus
        yield seed

However, the problem does not mention the seed value, nor do I know how to tell if it will achieve a full period. Any guidance would be appreciated

Answer 1

What is the seed?

The variable seed refers to the state of the RNG (Random-Number Generator). You must have some sort of changeable state, even if it's the position in a file or true random numbers, or the RNG will return the same number on every call.

For a LCG, the seed is simply the most recent value. You save that in a driver routine, or as some sort of internal status integer. It's usually an optional parameter; you provide it on the first call to "seed", or begin the sequence.

How to tell whether I get the full sequence?

Since the LCG state is entirely described by the most recent value, it returns a constant cycle of values, repeating that cycle for as long as you use it. To test this, you can simply start with a seed of 0 (any value will do), and then count how many iterations it takes to get that value again.

You can also determine the cycle length from the LCG parameters. This is a research project for the student (start with Wikipedia).

Does that unblock your worries?