## Lattice Points

A point $(x,y)$ with integral coordinates is called a lattice point. For example, $(\textrm{-1},0)$ is a lattice point, whereas $(3,1/2)$ is not. How many lattice points lie on the graph of $x^2+y^2=25$?
Source: NCTM Mathematics Teacher, January 2006

Solution
The graph of $x^2+y^2=25$ is a circle with center at $(0,0)$ and radius $5$. Any lattice point $(x,y)$ on the circle must form a Pythagorean triple with the radius. Since $(3,4,5)$ is a Pythagorean triple, $(3,4)$ is a lattice point on the circle.

From this initial lattice point we derive seven other lattice points by a series of reflections across the $xy$ axes for a total of eight.
Quadrant I: $(3,4),(4,3)$
Quadrant II: $(\textrm{-}3,4),(\textrm{-}4,3)$
Quadrant III: $(\textrm{-}4,\textrm{-}3),(\textrm{-}3,\textrm{-}4)$
Quadrant IV: $(3,\textrm{-}4),(4,\textrm{-}3)$

Plus the four special lattice points $(0,5),(0,\textrm{-}5),(5,0), and (\textrm{-}5,0)$.
$12$ lattice points lie on the graph of $x^2+y^2=25$.

Answer: $12$