On the representation of N by the powers of the golden mean and Zeckendorf theorem

TL;DR

This paper explores a novel representation of positive integers using powers of the golden ratio, extending Zeckendorf's theorem, and provides explicit formulas based on parity and non-positive powers.

Contribution

It introduces a new method to determine positive integers from non-positive powers of the golden ratio and their parity, expanding on Zeckendorf's theorem.

Findings

Explicit formula for N in terms of non-positive powers and parity

Extension of Zeckendorf's theorem to golden ratio powers

Unique representation of integers using this new method

Abstract

It is well known that every positive integer N can be written as the sum of non-consecutive powers of the golden ratio. We prove that the non-positive powers, together with the parity of the first positive power, can determine the positive integer. We give an explicit formula for N in terms of the above information.