0 1 1 2 3 5 8 13 . . .
Any mathematics enthusiast could immediately recognize this series within a blink of the eye.
The topic of discussion however is not limited to the Fibonacci Series (FS) alone, but beyond.
Not many would be aware of the Great Mathematician Édouard Lucas who studied the FS extensively.
He came out with another series which is closely related and integral to the FS.
This is known as the Lucas Series (LS). The sequence goes like this-
2 1 3 4 7 11 18 29 . . .
Now we might question as to why the Lucas series has to start with the number 2?.
What came to Lucas’s mind – no one is aware but we would be amazed to see that this series has tremendous use in the field of Number theory. When I delved deeper, I was amazed to understand that how integral were both FS and LS to the Golden Ratio Phi.
Any series starting with two random numbers and which would continue with the next term being the ‘sum’ of the previous two numbers would naturally become a series ‘similar’ to FS or LS.
When I say similar, this means that the ratio between two consecutive terms would be closer to the GR as the series approaches infinity.
Lets us see this as an example in the tabular for below.
The time taken by a particular series to reach the GR may vary, but slowly but surely it will reach there.
So, what makes the FS and LS unique?. There are a bunch of resources online covering this topic.
My interest here is to explore one aspect which I in a moment of epiphany discovered. This is discussed below.
The Golden Compound Number
If we happen to form a ‘compound’ number using numbers in FS and LS – we have an unique series of these compound numbers which I would like to call as the Fibonacci-Lucas Series (FLS).
Here, we use the Fibonacci number as the portion of the surd with sqrt(5) and the Lucas number without the sqrt(5) as an integer.
This concept is similar to how we express a complex number with the ‘real’ and ‘imaginary’ part.
Here in FLS, the compound numbers have the the ‘real’ and the ‘Golden’ part.
The unique aspect of this FLS is that the ratio of the consecutive terms is always the GR.
As discussed earlier, all series which has the next term as the sum of previous two terms would GR as the ratio, this series is special as the ratio of the ‘first two numbers’ would instantly give us the GR.
It is fascinating to note that the this ratio is the one which is known naturally as “THE GOLDEN RATIO” i.e. (1+ sqrt(5))/2.
This is valid on the negative side of the number line as well.
This is tabulated below for better clarity.
We can extend this to the family of ‘metal ratios’ like silver ratio and so on.
An example with the Pell’s series (PS) and the Pell-Lucas series (PLS) is also tabulated below.
Note that the new compound number series which we can call as the Pell+Pell-Lucas Series (PLPS) would have the Silver ratio as the ratio between two consecutive terms of the series.
The exploration of various series of similar nature is going on and we might have more discoveries soon.
Further properties of FLS and PPLS and new series would be explored in my future blog post.
Note : The pdf files of the above tables can be downloaded from the below link for study and analysis.
srikanthdinamani 