If we take the linear congruential algorithm with an additive component of 0,

Then it can be shown that if m is prime and if a given value of produces the maximum period of m - 1, then a^k will also produce the maximum period, provided that k is less than m and that and m - 1 are relatively prime. Demonstrate this by using X₀ = 1 and m = 31 and producing the sequences for ak = 3 3², 3³, and 3⁴.

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Xn+1 = (aXn) mod m