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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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