Skip to main content
4 of 6
added 96 characters in body; edited title
Jyrki Lahtonen
  • 133.9k
  • 29
  • 283
  • 680

A hyperbola passing through integer lattice points

Prove that for any $n\geq 0$, there is a hyperbola that passes through exactly $n$ lattice points and find an example.

For example it is easy to see that the hyperbola $xy=1$ passes throught exactly two lattice points. It is also easy to see that the hyperbola $$ x^2-2y^2=1 $$ passes through infinitely many lattice points, because this is a Pell equation and its integer solutions are related to the units of the ring of algebraic integers. The intermediate case of a prescribed number of lattice points requires another idea.

Kim Sokun
  • 117
  • 1
  • 2