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.
image
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)

image
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

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About mvtrinh

Retired high school math teacher.
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