Let be the set of all integers such that . For example, is the set . How many of the sets do not contain a perfect square?
For n≤50, the difference between n^2 and (n-1)^2 is less than 100. Since 50^2=2500, it follows that all the sets S_i for i≤25 will contain at least one square. For n>50, each square will differ from the previous one by more than 100, so each square will occupy a different S_i. The largest square less than 100,000 is 316^2=99,856. Thus the total number of sets S_i that do contain one or more squares is 26 (for the sets S_0 to S_25 inclusive) plus 266 (each containing exactly one of the squares from 51^2 to 316^2). Since 26+266=292, and there are altogether 1000 sets S_i (as i goes from 0 to 999), that leaves 1000–292=708 sets that do not contain a perfect square.