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\(k\)th-power residue chains. (English) Zbl 0361.10006
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

MSC:
11A15 Power residues, reciprocity
11B13 Additive bases, including sumsets
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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
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