Web site developers - 542 ANSWERS TO EXERCISES 3.3.3 ,l, 1) y=x-;
542 ANSWERS TO EXERCISES 3.3.3 ,l, 1) y=x-; Y++;-; / y=r-a! / l,l-a) yEz-I-a 2 2 M= g -cy 2 / / I,;-; > 1, c9 Fig. A-3. Permutation regions for a generator with potency 2; (Y = (u -l)c/m. 28. Fig. A-3 shows the various regions in the general case. The 213 region means UZ < UI < Us, if WI and UZ are chosen at random; the 321 region means that Us < U2 < Ul, etc. The probabilities for 123 and 321 are $ -cy/2 + ~ 12; the probabilities for all other cases are Q + (r/4 -cu2/4. To have all equal to &, we must have 1 -6cu + 6cu2 = 0. [This exercise establishes a theorem due to J. N. Franklin, Math. Comp. 17 (1963), 28-59, Theorem 13; other results of Franklin s paper are related to exercises 22 and 23.1 SECTION 3.3.4 1. VI is always m and ~1 = 2, for generators of maximum period. 2. Let V be the matrix whose rows are VI, . . . , vt. To minimize Y . Y, subject to the condition that Y # (0,. . . , 0) and VY is an integer column vector X, is equivalent to minimizing (V-lx). (V-lx), subject to the condition that X is a nonzero integer column vector. The columns of V- are UI, . . . , Ut.