Combinatorially fruitful properties of 3⋅2−1 and 3⋅2−2 modulo p


AbstractWrite a≡3⋅2−1 and b≡3⋅2−2(modp) where p is an odd prime. Let c be a value that is congruent (modp) to either a or b. For any x from Zp∖{0}, evaluate each of x and cx(modp) within the interval (0,p). Then consider the quantity μc∗(x)=min(cx−x,x−cx) where the differences are evaluated (modp−1, not modp) in the interval (0,p−1), and the quantity μc∧(x)=min(cx−x,x−cx) where the differences are evaluated (modp+1) in the interval (0,p+1). As x varies over Zp∖{0}, the values of each of μc∗(x) and μc∧(x) give exactly two occurrences of nearly every member of 1,2,…,(p−1)/2. This fact enables a and b to be used in constructing some terraces for Zp−1 and Zp+1 from segments of elements that are themselves initially evaluated in Zp

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Last time updated on 6/5/2019

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