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On the congruence \(2^{n-k}\equiv 1\) (mod n). (English) Zbl 0588.10002
A. Rotkiewicz [ibid. 43, 271-272 (1984; Zbl 0542.10003)] has recently proved that the congruence \(2^{n-2}\equiv 1 (mod n)\) has infinitely many solutions. Using a generalization of a result of Malo [see L. E. Dickson, History of the theory of numbers, Vol. 1, p. 93 (1971)], the following extension is immediate: Theorem. Each of the congruences \(2^{n-k_ i}\equiv 1 (mod n)\) \((i=0,1,2,...)\), where \(k_ 0=2\), \(k_{i+1}=2^{k_ i}-1\), has infinitely many solutions n. It remains an open question whether the congruence \(2^{n-k}\equiv 1 (mod n)\) has infinitely many solutions n for all positive integers k.

MSC:
11A15 Power residues, reciprocity
Keywords:
power residues
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