A point $$(x, y)$$ in the plane is called a lattice point if both its coordinates $$x, y$$ are integers. The number of lattice points that lie on the circle with center at $$(199, 0)$$ and radius 199 is
Solution
Writing $$u = x - 199$$, the lattice points satisfy $$u^2 + y^2 = 199^2$$. Since 199 is a prime of the form $$4k+3$$, it is not a sum of two nonzero squares and neither is $$199^2$$, so the only solutions have $$u = 0$$ or $$y = 0$$. These give the four points $$(199, \pm 199)$$, $$(0, 0)$$ and $$(398, 0)$$, so the count is 4.
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