Example 3 LCG (5, 0, 37, 1)

We obtain the sequence
1,5,25,14,33,17,11,18,16,6,30,2,10,13,28,29,34,22, 36,32,12,23,4,20,26,19,21,31,7,35,27,24,9,8,3,15, Table 3 provides the sequence throughout the period. Here, because we use a prime number as the divisor for the modulus operation and c = 0, we obtain a period one less than modulus 37 (0 is not possible, as it maps to itself, so we obtain a period of 36). Indeed, when m = p, a prime, the maximum period, , is m-1, even if . Thus, for linear, congruential generators with a prime modulus, using a non-zero c does not increase the period.

Here, the low order bits, while not exhibiting a discernible pattern, do not appear as ``random'' as one might expect. Indeed, as is shown in Altman [1], the bitwise randomness properties of LCGs should be considered on a case by case basis. He provides examples of LCGs with prime moduli that fail bitwise testing, but points out, for example, that does pass the bitwise randomness test.

(See exercise 8.)