They applied the QFT. After that, we measured the probability for some y's that we call good for some vectors that we call good and here's this probability for any of these good values, good vectors. Now, why do we care about this? Let's recall our definition of goodness, and let's write it without this absolute value here. So, 2Pi_Ary modular N divided by N is less or equal than Pi and greater than or equal N minus Pi. We're going to divide everything in this double inequality by this, by 2Pi A and multiplied by this, by N. You remember that A divided by N is approximately one divided by r. So, what we have here is minus r divided by two less or equal r_y modular N less or equal r divided by two. Now, we are going to get rid of this modular N. So, we can write ry is less or equal than NK plus r divided by 2 and greater or equal than NK minus r divided by 2. What we can see here is that for any particular K, we can solve this double inequality and find particular value y. So, when K equals to zero, you have y_0 which itself equals to zero. For K equal to one, I get some y_1, K equals to two, I get y_2, etc. When K equals to r, we have y_r has to be equal to N which is zero modular N and we have to start over. So, there are only r different good y's. We have the probability of measuring each of them. So, the probability of measuring any good y is this probability of measuring one of them multiplied by the number of good y's which is four divided by Pi squared, and it is almost one half. So, we see that it's reasonably good probability after the QFT, the measure some y's that we call good. Now, it's time to explain why we call all of them. So, all of them, we call them good. Why do we care about measuring these particular vectors? Okay. It's time to unveil the reason why we call this y's good. So, this is one of the definitions of goodness and we're going to divide it by N and by r. Okay. So, this divided by N, and by r we get y divided by N is less or equal. This would be K divided by r plus one by 2N. Here, we will have K divided by r minus one by 2N. That means that very near this number which we can compute because we have measured y and we have N. There is a number K divided by r and the distance between them is less than this, so it's very short distance. So, we have this inequality saying that near this number we can measure there is this number which is interesting to us because of this r. If r is less than square root of N, then here in this distance from y divided by N can be only one number with this quality. Why? Let's see why. If you have, for example, two denominators, r_1 and r_2, each such as r_1 less than square root of N and r_2 less than square root of N, and we have two fractions, a divided by r and b divided by r_2. What can be the distance between these numbers? We assume that these numbers are not equal. So, we compute the distance and equals to ar_2 minus br_1 divided by r_1, r_2. Since these fractions are not equal to each other and a and b are integers, let us read or equal to one divided by r_1, r_2. This is greater or equal than one divided by N because we have this inequalities for r_1 and r_2. So, if we have two fractions with the denominators which are less than square root of N, then the distance between them must be greater than this. So, if r happens to be less than square root of N, which is likely because N is quite a big number, then this value K divided by r is the only value with this quality in this neighborhood of y divided by N. So, we have y divided by N. We have this neighborhood or the size one divided by N and we can search in this neighborhood for the fractions with the denominators less than the square root. If you find one and if y was good y, then we get this fraction K divided by r, and we either get r or one of its denominators. But, how to search for this number which is close enough to this measured number with this small denominator by the continuous fraction method. Let's see an example of this.