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$N = \\Omega(\\pi(M))$ and gives a bound which is sharp up to the second order term, as Pach and S\\'andor gave an example for which\n  $$\n  N < \\pi(M)+ O\\left(\\frac {M^{2/3}}{\\log^2 M} \\right).\n  $$\n  The main new tool is a reduction of the original problem to the question of approximate 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