I am excited. This is my first geek-blog (for want of a better term, my apologies). I have always wanted to write something technical but try as I may, words elude me when I sit down to it. Unwillingly, I have had to acknowledge the tag given to me by the men in my family (sigh): technically-challenged.
It is not very hard for me to live up to the tag. So, when Ramana sent me his post this morning, I had been looking forward to it, albeit a bit anxious. I only had to open it to jump up in excitement. Mathematics! I almost clapped my hands with glee. It used to be my favourite subject in school, along with Literature.
The guest posts just keep getting better and better...
A few minor problems from number theory keep me awake in the nights from time to time. Roughly a month ago, for the price of a good night sleep, one of those nights of mental fecundity granted me an insight into the closed form equation for Fibonacci numbers. When Sangeetha asked me to write for her blog, I enthusiastically welcomed the opportunity since I meant to write about my insight for a while and I did not do so.
Fibonacci numbers, since I can grasp them even with my undernourished mathematical faculties, fascinate me days on end. In fact, I wrote a post on Fibonacci numbers more than one year ago at http://ramana.posterous.com/generalized-fibonacci-sequence. For any problem, depending on the dimension, there can be several generalizations! This time, my attention is the closed form expression for Fibonacci numbers also known as Binet's Fibonacci number formula. Of course, I did not know who Binet was until yesterday. In any event, in order to find out Nth term of Fibonacci sequence, we have the following formula:
F(N) = ((1+sqrt(5))^N - (1-sqrt(5))^N)/(2^Nsqrt(5))
The above formula amazed me for many years since I did not know much about "Linear Recurrence Equation" (http://mathworld.wolfram.com/LinearRecurrenceEquation.html). Yes, I would love to become a mathematician and that too in number theory without knowing Linear Recurrence Equation and Binet's work.
Let us get back to the above equation and the night of pure ecstasy. I expanded the above formula for a few values of N. I recalled the following fact about binomial expansion:
(a+b)^N - (a-b)^N = 2(N c 1)a^(N-1)b + 2(N c 3)a^(N-3)b^3 + ...
As you could see in the above, the powers of b are odd numbers. In our closed formula for Fibonacci, a equals 1 and b equals sqrt(5). Since the powers of b (sqrt(5)) are odd, we can get rid of one of sqrt(5) with the one in the denominator. Even powers of sqrt(5) result in nice integers. I was extremely delighted to note this. Of course, simple minds and simple pleasures. Then, it occurred to me why we cannot change the value of b in the closed formula. That is what I did next. I substituted sqrt(7) for sqrt(5). I ended up getting the following sequence:
1, 1, 2.5, 4, 7.75, 13.75, ...
If you examine the above for a few seconds, it will reveal its recurrence relation:
g(N) = g(N-1) + 1.5g(N-2)
From the above result, as you can easily surmise, I was totally ecstatic. Then, I tried sqrt(9) equal to 3. I got:
1, 1, 3, 5, 11, 21, 43, 85, 171, ...
The relation is:
h(N) = h(n-1) + 2h(n-2)
Now, it is time to mess with the value of a. Until now, it has been 1. I increased it 2 and I still kept b at 3. I got:
1, 2, 5.25, 13, 32.5625, ...
The relation is:
j(N) = 2j(N-1) + 1.25j(N-2)
The above results are the ones that I wanted to post in this entry. Since it is Sangeetha's blog, I decided to check a couple of web sites before posting. Unfortunately, I discovered this page http://mathworld.wolfram.com/FibonacciNumber.html. The cases that I was discussing above are straight from the following equation:
X^2 - aX - b = 0
The moral of the story comes in a twin-pack:
1. You cannot aspire to be a mathematician and accomplish "original" discoveries without studying even the basics.
2. At the same time, there is no shame in exploring mathematical world even with limited knowledge. As long as you do not expect and ask for huge rewards for your investigations, you can have a lot of fun. In addition to that, the discoveries that you make are undoubtedly original for you if not for the rest of the world.
Literature stayed with me in the form of the books I read, this blog, emails I write everyday, more recently Twitter, et al. Somewhere along the road, I had left Math behind. I can't wait to start clicking on those links!