tion exhibits four equiprobable peaks (Pmax = 0.25) at the register locations k = 0, 512, 1024, 1536, with any other measurement probabilities being zero. This 20.6 Factorizing N = 15 and other nontrivial composites probability distribution represents the ideal measurement conditions with zero failure. As we have learnt, the actual measurement is successful only within some error proba- bility E, which is a function of the register size K, and which has been set in this example to E = 0.25. To recall, this failure probability can be made as small as desired, hence making the measurement reality closer to the ideal probability distribution shown in Fig. 20.5. The existence of four equiprobable probability peaks betrays the periodicity of the function x n mod N, which is of the order r = 4. But the experimentalist has no precon- ceived idea of this, and is allowed to make only a few measurement attempts. Then it is equally likely for each of the measurements to hit the values ki = 0, 512, 1024, or 1536. To recall, these measurements correspond to the phase =s/r, with 1 being defined in K bits. The measurement, thus, corresponds to the rational numbers ki/M = ki/2K ki/213 = ki/2048, corresponding to the four possible determinations The first determination, ki = 0, does not give any clue as to the order r. Measur- ing it is, therefore, a failed attempt. Consider the next two other determinations, i.e., ki = 512, 1024. We have k1 = 512/2048 = 1/4 and k2 = 1024/2048 = 1/2. These two rational numbers do not meet the condition in Eq. (20.25) since (1,4) and (1, 2) are not co-prime. Therefore, they cannot be convergent on Q. The last possible measurement, k3 = 1536 lends itself to the continued fraction expansion: which, according to the algorithm (see Appendix R) gives the series of fractions po/qo = 0/1, p1/91 = 1/1, and p2/92 = 3/4. Since (3,4) are co-prime, the fraction 3/4 is a convergent of Q and thus r = 92 = 4 is of the order of X. Luckily, r = 4 is even, and, furthermore, we have x r / 2 mod N = 82 mod 15 = 4 mod