Given a hyperbola $$y = x + \frac n x \tag 1$$ and a range $x \in (a,b)$, how do we find the number of lattice points lying in the range that the hyperbola passes through? A lattice point is a point $(x,y)$ on the coordinate plane with both $x$ and $y$ integers.
One approach is to factor $n$ and then take $x$ as the divisors of $n$ that lie in the range $(a,b)$.
Example: Find lattice points of $$y = x + \frac {527} x \tag 2$$ in the range $x \in (0,20)$. We factor $527 = 17 \times 31$. The divisors are $1, 17, 31$ and $527$ and only $x \in \{1, 17\}$ are in the range $(0,20)$. Therefore, they are the $x$-coordinates of the lattice points of interest. We substitute in Eqn. (2) to get $(1, 528), (17, 48)$ as the lattice points.
Question 1: This method works when $n$ is of a manageable size and can be factored easily. What do we do when $n$ is large?
Question 2: Even if we are not able to find the exact lattice points, is there a way to
- estimate the number of solutions in the range (or)
- even just the existence of solutions in the range (i.e., yes/no answer for the existence question)
Assume that we are given the search range $x \in (a,b)$.