Hello, prasum!
The number of integral points on the hyperbola is __.
We have: .
n . .
As ASZ pointed out,
. . which has 32 pairs of factors.
. .
We have: .
Hence: .
Then: .
Each of the 32 pairs of factors will provide us with a solution.
We see that: .
So each of these factorings gives us four solutions.
Except the last pair
. . which produces only two solutions: .
Therefore, there are: . lattice points on the hyperbola.
In Soroban's solution, you need to exclude the seven factorisations of the form , where one factor is even and the other factor is odd. Those will not lead to integer solutions for x and y. Thus the correct answer is that the number of integral points on the hyperbola is