 for which xr/2 -1 mod N. and mod N}. The results for N = 15, 21, 33, 35, 39 are shown in Fig. 20.6. The darkened data correspond to odd values of r, while the shaded data correspond to r values for mod N. The graph, thus, makes it possible to calculate the probabilities each subset of x satisfying the algorithm requirements, and the indicator In the case N = 21, for instance, we find = {2, 10, 11, 19} for r = 6 and x = {8, 13} for r=2, corresponding to p{1 = 4/18 2/9 and p2 2/18= 1/9, respectively. The associated values of r give P2 =1/3 and (1) pl2== respectively, yielding II = (2/9) X (1/3) + 0 = 2/27 = 7.4%. Considering next the case N = 33, we find X = {5, 7, 13, 14, 19, 20, 26, 28} for = 10,x {10,21,23} for r = 2, corresponding 4/15 and pl2 3/30 1/10, respectively. For = 10, the probability distribution exhibits 10 peaks at the k locations: 0123456789 10' 10' 10' 10' 10' 10 10' 10' 10' corresponding to the irreducible fractions p/q 0113213749 10'10' 5' 10' 5' 2' 5' 10' 5' 10' in which only six have p,q as co-prime numbers. We, thus, have pl = 6/10 = 3/5 and also pl2 0, yielding II = (4/15) X (3/5) + 0 = 4/25= 16.0%. 20.6 Factorizing N = 15 and other nontrivial composites y = 0.14In(x) - 0.306 Composite N Figure 20.7 Indicator function I((N), corresponding to the probability of successfully factoring N in a single run of Shor's algorithm. Similar analysis of the other N values leads to the indicator function II(N). To recall, such a function represents the probability of successfully factorizing N based on a single random selection of x E [2, N - 2]. The function II(N) is plotted in Fig. 20.7 for N < 100. We observe from the plot that II I(N) increases with N accord- ing to a logarithmic fit, along with some scattering. Such scattering is explained by either "lucky" or "unlucky" composites. For instance, N = 15 and N = 51 are lucky in the sense that all periods r satisfy the two conditions of being even and such that xr/2 # 1 mod N for any x E [2, N - 1]. This property maximizes the probability p(1). In cont