Patterns of quadratic residues and nonresidues for infinitely many primes

被引:13
作者
Wright, Steve [1 ]
机构
[1] Oakland Univ, Dept Math & Stat, Rochester, MI 48309 USA
关键词
quadratic residue; quadratic nonresidue; Galois field; recurrence relation;
D O I
10.1016/j.jnt.2006.06.003
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
If S is a nonempty, finite subset of the positive integers, we address the question of when the elements of S consist of various mixtures of quadratic residues and nonresidues for infinitely many primes. We are concerned in particular with the problem of characterizing those subsets of integers that consist entirely of either (1) quadratic residues or (2) quadratic nonresidues for such a set of primes. We solve problem (1) and we show that problem (2) is equivalent to a purely combinatorial problem concerning families of subsets of a finite set. For sets S of (essentially) small cardinality, we solve problem (2). Related results and some associated enumerative combinatorics are also discussed. (c) 2006 Elsevier Inc. All rights reserved.
引用
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页码:120 / 132
页数:13
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