(In reply to
I think ... by Larry)
Looking at positive x and its associated integer N: For any interval [sqrt(N), sqrt(N+1)) the left side is constant but the right side is continuously increasing so there can only be at most one possible solution.
So lets take your found root x=sqrt(N)+1/50. This will be the intersection, so long as it is still within the interval. Then we need sqrt(N)+1/50 < sqrt(N+1). Square each side and simplify and we get sqrt(N) < 24.99, or N < 624.5001.
So then there are 625 integer values of N that work according to your formula: N=0 to 624. Charlie's brute force search matches this range.
re: I think ...
