Christian L. Yankov

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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 Gn = {(a, b) : a, b ∈ Z, ab ≡ 1 (mod n), 1 ≤ a, b ≤ n − 1} . 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(More)
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 arguments(More)
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