Editing Prime modulus multiplicative linear congruential generator


:<math>y_{n+1}\equiv ay_n + b~~~(\mod ~m),</math>

The parameter <math>m</math>
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 <math>m</math> is the Mersenne prime

:<math>\left.m\right.=2^{31} -1=2147483647</math>,

with  <math>a=7^5</math> (a positive primitive root of <math>m</math> see Ref.s 1 and 2),
and <math>b=0</math>.
With these parameters one is able to generate a series of
<math>2.147 \times 10^9</math>
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 <math>m</math> and <math>a</math> see Ref.4
and for its use on 64-bit computers see Ref. 5.

==References==
#[CACM_1966_09_0432]
#[IBM_1969_02_0136]
#[CACM_2003_46_0090]
#[MC_1999_68_249]
#[CPC_1997_103_0103]
[[Category: Random numbers]]
@@ Line 1: / Line 1: @@
-The '''prime modulus multiplicative linear congruential generator''' is a special type of
-[[linear congruential generator]], given by:
+:<math>y_{n+1}\equiv ay_n + b~~~(\mod ~m),</math>
-:<math>y_{n+1}\equiv ay_n + b  \qquad({\mathrm {mod}} ~m),</math>
 The parameter <math>m</math>
@@ Line 21: / Line 20: @@
 ==References==
-#[http://dx.doi.org/10.1145/365696.365712 D. W. Hutchinson, "A New Uniform Pesudorandom Number Generator", Communications of the ACM, '''9''' pp. 432-433 (1966)]
+#[CACM_1966_09_0432]
-#[http://www.research.ibm.com/journal/sj/082/ibmsj0802D.pdf P. A. W.  Lewis and A. S. Goodman and J. M. Miller, "A pseudo-random number generator for the System/360", IBM Systems Journal  '''2''' pp. 136 (1969)]
+#[IBM_1969_02_0136]
-#[http://dx.doi.org/10.1145/769800.769827 G. Marsaglia, "Seeds for Random Number Generators",Communications of the ACM, '''46''' pp. 90-93 (2003)]
+#[CACM_2003_46_0090]
-#[http://dx.doi.org/10.1090/S0025-5718-99-00996-5 Pierre L'Ecuyer, "Tables of linear congruential generators of different sizes and good lattice structure", Mathematics of Computation, '''68''' pp. 249 - 260  (1999)]
+#[MC_1999_68_249]
-#[http://dx.doi.org/10.1016/S0010-4655(97)00052-0   Iosif G. Dyadkin and Kenneth G. Hamilton "A study of 64-bit multipliers for Lehmer pseudorandom number generators", Computer Physics Communications, '''103'''  pp. 103-130 (1997)]
+#[CPC_1997_103_0103]
-#[http://dx.doi.org/10.1016/S0010-4655(99)00467-1    Iosif G. Dyadkin and Kenneth G. Hamilton "A study of 128-bit multipliers for congruential pseudorandom number generators", Computer Physics Communications, '''125'''   pp. 239-258 (2000)]
 [[Category: Random numbers]]