Expressing Numbers in terms of Golden, Silver and Bronze Ratios

The idea of expressing certain kind of numbers as linear combination of special class of numbers has always been an interesting exercise in mathematics. In this paper, I present an interesting way to write a given natural number as sum or difference of integral powers of golden ratio, silver ratio and bronze ratio. Suitable illustrations enabling the process are briefed in the paper.


Introduction
Leonardo Fibonacci of Italy published his famous book Liber Abaci in 1202 CE. This book was instrumental in introducing Indian numerals to entire world. Apart from various interesting problems it possessed, the problem on growth of immortal rabbits paved way for describing the most famous Fibonacci numbers, the sequence of numbers named after him. A slight modification of Fibonacci sequence leads to a sequence called Lucas Sequence. It is well known that the ratio of successive Lucas numbers approaches to Golden Ratio, which has so much of significance not only in mathematics but also in Architecture and Engineering. In this paper, I will prove some important results for expressing any natural number as sum or difference of integral powers of gold, silver and bronze ratios. where 01 2, 1 LL == . Using (2.1), the terms of the Lucas sequence are given by 2,1, 3,4,7,11,18,29,47,76,123,199,… We observe that except the first two terms, each term is sum of two preceding terms.

Integral Powers of Golden Ratio
In the following theorem, I will relate sum of integral powers of golden ratios to Lucas numbers.  Now by Induction Hypothesis, we will assume that (3.1) holds true for all values of n up to some natural number k and we will prove the result for n = k + 1.
Thus, we assume that Thus the result is true for n = k + 1 whenever it is true up to n = k. Hence, by Induction Principle, equation (3.1) is true for all natural numbers n. This completes the proof.

Theorem 2
Any natural number can be expressible as sum or difference of integral powers of golden ratio. Proof: By Zeckendorf's theorem we know that any natural number can be expressed uniquely as a sum of non-consecutive Lucas numbers. By (3.1), we know that for any natural number n, the nth Lucas number is expressible as nn  − − if n is odd and nn  − + if n is even. Hence any natural number can be expressible as sum or difference of integral powers of golden ratio. This completes the proof.

Illustrations
In this section, I will demonstrate the technique of actually expressing any natural number as sum or difference of integral powers of golden ratio (integer powers of golden ratio).
If we consider the number 12, the number of months in any year, then first we will try to express 12 as sum of terms of Lucas sequence. Doing so, we get 12 = 1 + 11. Now using (3.1), we have Similarly, for 1729 we have, . Similar to these illustrations, given any natural number, first we try to express that sum of Lucas numbers and using (3.1), we can express as sum or difference of integral powers of golden ratio.
In the following theorem, I will relate sum of integral powers of silver ratios to the terms of the sequence defined in 2. By Induction Hypothesis, we will assume that (4.1) holds true for all values of n up to some natural number k and we will prove the result for n = k + 1.
Thus, we assume that  Thus if the result is true for all values up to k, then it is also true for k + 1. Hence by Induction Principle, (4.1) is true for all natural numbers n. This completes the proof.

Theorem 4
Any natural number can be expressible as sum or difference of integral powers of silver ratio.
Proof: First, we note that 1 can be written as integral power of silver ratio in the form

Illustrations
In this section, I will consider four natural numbers greater than 1 and try to express them as sum or difference of integral powers of silver ratio. The terms of the sequence corresponding to (2.3) are given by  1,2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,228486,… If we consider 12, then using the terms of the sequence defined in (2.2) and (4.1), we have These illustrations verify theorem 4 established above.

Integral Powers of Bronze Ratio
In the following theorem, I will relate sum of integral powers of bronze ratios to the terms of the sequence defined in 2.  Thus if the result is true for all values up to k, then it is also true for k + 1. Hence by Induction Principle, (5.1) is true for all natural numbers n. This completes the proof.

Theorem 6
Any natural number can be expressible as sum or difference of integral powers of bronze ratio with two repetitions of first term 1 of the sequence defined in (2.3) Proof: First, we note that 1 can be written as integral power of bronze ratio in the form

Illustrations
In this section, I will consider two natural numbers and try to express them as sum or difference of integral powers of bronze ratio.
The terms of the sequence corresponding to (2.3) are given by  1,3,11,36,119,393,1298,4287,17148, In this case, we notice that the constant term 1 = 0  gets repeated twice.
These illustrations verify theorem 6 established above.

Conclusion
The concept of expressing any natural number as sum or difference of integer powers of golden, silver and bronze ratios has been proved in this paper. This has been done using theorems 2,4 and 6 respectively. While for golden and silver ratios, there is no need for repetition of any terms of the corresponding sequences considered, for bronze ratio, in some cases we require the repletion of the first term namely 1 twice to achieve the decomposition of the given natural number as sum or difference of integer powers of bronze ratio.
We can extend this process for other metallic ratios also but in that case we need more repetitions of the terms of the sequence describing them, since the terms of general metallic ratio sequence gets widely distributed. Hence, I just considered the first three metallic ratios in this paper.