Prime Numbers

398 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
8.2.1 Modular methods
The veritable workhorse of the random number generation industry has been
the linear-congruential generator. This method uses an integer iteration
xn+1 =(axn + b) modm,
where a, b, m are integer constants with m > 1, which recursion is to be
ignited by an initial “seed,” say x0. To this day there continue to appear
research results on the efficacy of this and related generators. One variant is
the multiplicative congruential generator, with recursion
xn+1 =(cxn) modm,
where in this case the seed x0 is assumed coprime to m. In applications
requiring a random() function that returns samples out of the real interval
[0, 1), the usual expedient is simply to use xn/m.
Recurrences, like the two above, are eventually periodic. For random
number generation it is desirable to use a recursion of some long period. It is
easy to see that the linear-congruential generator has period at most m and
the multiplicative congruential generator has period at most m−1. The linear
case can—under certain constraints on the parameters—have the full period
m for the sequence (xn), while the multiplicative variety can have period
m − 1. Fundamental rules for the behavior of such generators are embodied
in the following theorem:
Theorem 8.2.1 (Lehmer). The linear-congruential generator determined
by
xn+1 =(axn + b) modm
has period m if and only if
(1) gcd(b, m) =1,
(2) p|a − 1 whenever prime p|m,
(3) 4|a − 1 if 4|m.
Furthermore, the multiplicative congruential generator determined by
has period m − 1 if and only if
(1) m is prime,
(2) c is a primitive root of m,
(3) x0 ≡ 0(modm).
xn+1 =(cxn) modm
Many computer systems still provide the linear scheme, even though there are
certain flaws, as we shall discuss.
First we give an explicit, standard linear-congruential generator: