On the representation of the natural numbers by powers of the golden mean

Michel Dekking, Ad Van Loon

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Abstract

In a base phi representation, a natural number is written as a sum of powers of the golden mean φ. There are many ways to do this. Well known is the standard representation, introduced by George Bergman in 1957, where a unique representation is obtained by requiring that no consecutive powers, φn and φn+1, occur in the representation. In this paper, we introduce a new representation by allowing that the powers φ0 and φ1 may occur at the same time, but no other consecutive powers. We then argue that this representation is much closer to the classical representation of the natural numbers by powers of an integer than Bergman’s standard representation.

Original languageEnglish
Pages (from-to)105-118
Number of pages14
JournalFibonacci Quarterly
Volume61
Issue number2
Publication statusPublished - 2023

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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