A point with integral coordinates is called a lattice point. For example, is a lattice point, whereas is not. How many lattice points lie on the graph of ?

Source: NCTM Mathematics Teacher, January 2006

**Solution**

The graph of is a circle with center at and radius . Any lattice point on the circle must form a Pythagorean triple with the radius. Since is a Pythagorean triple, 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 axes for a total of eight.

Quadrant I:

Quadrant II:

Quadrant III:

Quadrant IV:

Plus the four special lattice points .

lattice points lie on the graph of .

**Answer**:

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