Abstract
This paper considers the following questions. 1. (i) Are there always m “consecutive” kth power residues modulo a Gaussian prime within a certain distance of the origin with the distance independent of the Gaussian prime? 2. (ii) Are there always m “consecutive” elements in some coset with respect to the multiplicative subgroup of kth power residues modulo a Gaussian prime that are within a certain distance of the origin with the distance of the Gaussian prime? The answers are shown to be yes if m = 2,3 and k = 2,3 and no if k ≥ 2 and m ≥ 4. The problems are not solved for k > 3 and m = 2,3.
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