Geometric series representation of natural numbers by subtracting 2 from Fibonacci numbers

0. Introduction

In the previous article, we discussed the geometric series representation of natural numbers by subtracting one from the Fibonacci numbers (https://isiotoyume.wixsite.com/website/post/コラッツ予想の証明に関する概説). In this article, we will introduce the geometric series representation of natural numbers by subtracting 2 from the Fibonacci numbers.

1. Review

The Fibonacci numbers minus one can uniquely indicate the natural numbers as follows

(1) 5=7-2

(2) 79=88-(12-(4-1))

2. Newly discovered

The Fibonacci numbers minus 2 can be uniquely shown to be natural numbers as follows

(1) 5=6-1

(2) 79=87-(11-3)

3. geometric series representation of natural numbers

4. For the Fibonacci numbers minus n

The absolute value of the negative number of a Fibonacci number is equal to the value of the positive number. Also, the odd-numbered numbers will be negative and the even-numbered numbers will be positive.

Using this, we should be able to represent any natural number by the Fibonacci number minus n, perhaps for any natural number n.

At this point, if it is realized by addition, as in Zeckendorf's theorem (n=0,3,4,...) and the case realized by subtraction (n=1,2,...) as introduced above. It seems that we can categorize them into two types.

I wonder where this difference comes from?

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