Prime modulus multiplicative linear congruential generator

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{n+1}\equiv ay_n + b~~~(\mod ~m),}

The parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is the Mersenne prime

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.m\right.=2^{31} -1=2147483647} ,

with $a=7^{5}$ (a positive primitive root of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} see Ref.s 1 and 2), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=0} . With these parameters one is able to generate a series of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]