Vegh, Emanuel \(k\)th-power residue chains. (English) Zbl 0361.10006 J. Number Theory 9, 179-181 (1977). Summary: A finite sequence of integers for which all the sums of consecutive terms are distinct \(k\)-th power residues modulo \(p\) ( \(p\) a prime, \(k\) a positive integer), is called a chain of \(k\)-th power residues modulo \(p\). If moreover, any permutation of the sequence leads to a chain of \(k\)-th power residues modulo \(p\), the sequence is called a permutation chain of \(k\)-th power residues modulo \(p\). The author shows that given a prime \(k\) and a positive integer \(m\), the sequence \(\{1,2,2^2,\dots,2^{m-1}\}\) is a permutation chain of \(k\)-th power residues modulo \(p\) for infinitely many primes \(p\). The result is an immediate consequence of a special case of a result of Kummer concerning characters with preassigned values. However, using the result of W. H. Mills [Can. J. Math. 15, 169–171 (1963; Zbl 0125.02308)] instead of Kummer’s one, the result can be slightly extended to other values of \(k\). Reviewer: Štefan Porubský Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 Review MSC: 11A15 Power residues, reciprocity 11B13 Additive bases, including sumsets PDF BibTeX XML Cite \textit{E. Vegh}, J. Number Theory 9, 179--181 (1977; Zbl 0361.10006) Full Text: DOI References: [1] Mills, W.H., Characters with preassigned values, Canad. J. math., 15, 169-171, (1963) · Zbl 0125.02308 [2] Graham, R.L., On quadruples of consecutive k-th power residues, (), 196-197 · Zbl 0128.26802 [3] Gupta, H., Chains of quadratic residues, Math. comp., 25, 379-382, (1971) · Zbl 0218.10005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.