Period of a Linear congruential generator

Question

I found by experimentation that for the pseudorandom generator described below, the period is 32 (https://repl.it/repls/EasySphericalPhysics).

$X_0=0$

$X_{n+1}=(34 * X_n + 17) \bmod 97$

Although the Hull–Dobell Theorem conditions are fulfilled which should mean that the period equals 97.

• $17$ and $97$ are relative prime

• $34 - 1$ is divisible by 1 (prime factors of 97)

• $97$ is not divisible by 4

How can this be explained.

Answer 1

The second condition of the Hull-Dobell theorem requires that the multiplier less $1$ (i.e. $33$ in this case) be divisible by all prime factors of the modulus. This isn't satisfied here because there's a prime factor of the modulus (namely $97$) which doesn't divide $33$.