TITLE: Why do prime numbers in modulo result in more uniform distributions?
QUESTION [5 upvotes]: Let us assume a sequence as follows:
$S_{n} = (S_{n-1} * c_{1} + c_{2})\text{ mod } m$
This is the pseudorandom generator found in most programming languages' random function. 
It is known that a prime $m$ results in a more uniform distribution of random numbers, as a result of a larger period for $S_{n}$. As a result, $m$ is typically a prime number.
Why do prime numbers typically result in larger periods than factorable numbers for modulo arithmetic?

REPLY [2 votes]: According to Wikipedia, the period is at most $m$, and is equal to $m$ only if 

$\gcd(c_2,m)=1$, 
$p\mid m$ implies $p\mid c_1-1$ for all prime $p$, and 
$4\mid m$ implies $4\mid c_1-1$. 

So $m$ needn't be prime, but it's easiest to meet and to check these conditions if it is.