+1729 Let
f(x) = \sqrt{x - \sqrt{x}}
Find the largest three-digit value of x such that f(x) is an integer.
+9676 Suppose f(x) is an integer. Consequently sqrt(x) is an integer. Suppose N = sqrt(x). Then sqrt(x - sqrt(x)) = sqrt(N^2 - N) = sqrt(N(N-1)).
x is 3 digits, so the range of N is √100≤N≤√961, which is 10≤N≤31. There are no integer in this range where N(N - 1) is a square number.
Hence, there are no solution.
