Noncanonical number systems in the integers

Details Noncanonical number systems in the integers Noncanonical number systems in the integers

2008-04-14

arxiv.org/abs/0804.2190v1

Journal of Number Theory 128 (2008) 2914-2938

Mathematics

Number Theory

30 pages, to appear in Journal of Number Theory

Scientific paper

10.1016/j.jnt.2008.02.012

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b|\ge 4, and for b=-2. For the special case b=-2, we give a full characterisation of all valid digit sets.

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