Fibonacci Expansions and 'F-adic' Integers

Abstract

In this paper we first define the Fibonacci expansion, also known as the Zeckendorf expansion, of a positive integer as the unique expression of n as the sum of non-consecutive positive Fibonacci numbers. By analogy with p-adic integers, we then define F-adic integers as a generalization of positive integers in terms of their Fibonacci expansions, which are now allowed to be infinite. Finally we show that the F-adic integers are isomorphic to the circle group as a topological group, with ordinary addition as group multiplication.

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