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We investigate the distribution of n − M (n) where M (n) = max { |a − b| : 1 ≤ a, b ≤ n − 1 and ab ≡ 1 (mod n)}. Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on n − M (n). We also present some heuristic… (More)

In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set The heuristic is based on an asymptotic formula of Rényi and Sulanke. After describing two algorithms to determine the convex closure, we 1 compare the numeric results with the heuristic estimate. The numeric results do not… (More)

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