On unavoidable obstructions in Gaussian walks

2015/11/10 by Somu, Sai Teja, Pandey, Ram Krishna
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1511.03237

Abstract

In this paper we investigate a problem about certain walks in the ring of Gaussian integers. Let n,d be two natural numbers. Does there exist a sequence of Gaussian integers z_j such that |z_j+1-z_j|=1 and a pair of indices r and s, such that z_r-z_s=n and for all indices t and u, z_t-z_u≠ d? If there exists such a sequence we call n to be d avoidable. Let A_n be the set of all d∈ ℕ such that n is not d avoidable. Recently, Ledoan and Zaharescu proved that \d ∈ ℕ : d|n\⊂ A_n. We extend this result by giving a necessary and sufficient condition for d∈ A_n which answers a question posed by Ledoan and Zaharescu. We also find a precise formula for the cardinality of A_n and answer three other questions raised in the same paper.

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