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Inequality on number of lattice points

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October 20, 2010
inequalitiesanalytic geometrynumber theory unsolvednumber theory

Problem Statement

Given a natural number nn, prove that the number M(n)M(n) of points with integer coordinates inside the circle (O(0,0),n)(O(0, 0),\sqrt{n}) satisfies πn−5n+1<M(n)<πn+4n+1\pi n - 5\sqrt{n} + 1<M(n) < \pi n+ 4\sqrt{n} + 1
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