Prime modulus multiplicative linear congruential generator

$y_{n+1}\equiv ay_{n}+b~~~(\mod ~m),$

The parameter $m$ should be prime and as large as possible without causing a numerical overflow on the computer that it is running on. For example, for a 32-bit (31 bit + 1 sign bit) word size then the logical choice of $m$ is the Mersenne prime

$\left.m\right.=2^{31}-1=2147483647$,

with $a=7^{5}$ (a positive primitive root of $m$ see Ref.s 1 and 2), and $b=0$. With these parameters one is able to generate a series of $2.147\times 10^{9}$ pseudo-random numbers from one seed value. For an interesting discussion on how to choose an initial seed value see Ref. 3. For a list of other values of $m$ and $a$ see Ref.4 and for its use on 64-bit computers see Ref. 5.

References

1. [CACM_1966_09_0432]
2. [IBM_1969_02_0136]
3. [CACM_2003_46_0090]
4. [MC_1999_68_249]
5. [CPC_1997_103_0103]