Patterns of quadratic residues and nonresidues for infinitely many primes

JOURNAL OF NUMBER THEORY

卷:123

Article

Wright, Steve

关键词: quadratic residue;    quadratic nonresidue;    Galois field;    recurrence relation;   

DOI  :  10.1016/j.jnt.2006.06.003

来源: Elsevier

【 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.

【 授权许可】

Free