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Simple arguments on consecutive power residues

Zhi-Wei Sun
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Abstract

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii) Let O_K be the ring of algebraic integers in a quadratic field $K=Q(\sqrt d)$ with d in {-1,-2,-3,-7,-11}. Then, for any irreducible $π\in O_K$ and positive integer k not relatively prime to $π\barπ-1$, there exists a k-th power non-residue $ω\in O_K$ modulo $π$ such that $|ω|<\sqrt{|π|}+0.65$.

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Zhi-Wei Sun. 2007-02-22. Simple arguments on consecutive power residues. https://arxiv.org/abs/math/0312010

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