Simple arguments on consecutive power residues

✍ Scribed by Zhi-Wei Sun

• Book ID: 104024781
• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 101 KB
• Volume: 124
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.08.003

✦ Synopsis

By some extremely simple arguments, we point out the following:

(i) If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth 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( √ d ) with d ∈ {-1, -2, -3, -7, -11}. Then, for any irreducible π ∈ O K and positive integer k not relatively prime to π π -1, there exists a kth power non-residue ω ∈ O K modulo π such that |ω| < √ |π | + 0.65.