Patterns of quadratic residues and nonresidues for infinitely many primes
Patterns of quadratic and higher residues have been studied by many mathematicians from Edouard Lucas [see \textit{Anne-Marie Décaillot}, Rev. Hist. Math. 4, No. 2, 191--236 (1998; Zbl 0938.01035), as well as Rev. Hist. Math. 8, No. 2, 145--206 (2002; Zbl 1111.01004)] to the Lehmers [\textit{D. H. Lehmer} and \textit{E. Lehmer}, Proc. Am. Math. Soc. 13, 102--106 (1962; Zbl 0104.03704)]. While it is not difficult to see that there are infinitely many primes \(p\) for which a given pair \((x,x+1)\) is a pair of quadratic residues, it is clear that \((x,x+1)\) has no chance of being a pair of nonresidues if \(x\) or \(x+1\) is a square. The author uses combinatorial (and quite technical) conditions that characterize sets of at most four integers that support all patterns. Actually, this problem was solved in complete generality for all number fields by Kummer and Hilbert, using the analytic methods of Dirichlet and Dedekind's class number formula: Let \(K\) be a number field; if \(\alpha_1, \ldots, \alpha_n\) are elements of \(K\) that are independent modulo squares, then there exist infinitely many prime ideals \({\mathfrak p}\) such that the quadratic residue symbols \((\alpha_1/{\mathfrak p}), \ldots, (\alpha_n/{\mathfrak p})\) have prescribed values \(\pm 1\).
scientific article; zbMATH DE number 5129749
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