My Research Paper on M-Bonacci Sequences
My love for Mathematics really blossomed in the summer of 2017, when I first discovered the enigma that is Calculus. Or more precisely, Differential Calculus. The subject intrigued me for multiple reasons- its inherent ability to mesmerize its explorer while still being simple, its links with Algebra and Geometry, and most of all, the fact that it opens up beautiful avenues for the Math enthusiast and paves the way for a beautiful journey of discovery and enlightenment!
Now that’s a whole lot of generic talk. But it’s true nonetheless; from my high school classes that year onward, I became more interested in the discipline than I ever was. I would be engrossed by math-discussions during classes and once they got over, I would explore the generalizations of the problems we already discussed during class.
One of many such explorations was when I discovered how powerfully close the Fibonacci sequence (1,1,2,3,5,8,13,…) and its characteristic equation ( x²-x-1=0) are. I would find weird, objectively uninteresting (yet interesting for me) equations about this relationship and prove them after classes, and my Math teacher would repeatedly flash an uninterested/disinterested expression reminiscent of a purist’s reaction to the applications of his/her discovery.
My luck finally improved in the winter of the same year. My teacher would repeatedly ask me to find generalizations- not of the inelegant formulae I kept obtaining, but of the equation x²-x-1=0 itself. The idea wasn’t lucrative at first, but I only understood what he meant after a considerable while. During my winter vacation, I decided to spend a majority of time doing mathematical explorations, hoping they would lead to enlightening conjectures which I could prove later.
I decided to start with taking a higher-degree version of the quadratic (the characteristic equation). I took x³-x²-x-1=0 and did the same explorations I conducted with the quadratics.
I found some amazing generalizations. For example, the limit of the ratio of consecutive terms in the Fibonacci sequence (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, and so on) starts to approach the golden ratio once the terms get very large. This golden ration also happens to be the largest magnitude root of x²-x-1=0. Similarly, if you take the limit of the ratio of consecutive terms in the sequence 1,1,2,4,7,13,24,44,81,… you end up getting the largest magnitude real root of x³-x²-x-1=0!
Notice how the first (Fibonacci) sequence had each successive term being obtained after adding the previous two terms. The second sequence was obtained by adding the previous three terms. This was when I took the third degree version of my equation. I found out on various internet sources that such a sequence was called the Tribonacci sequence.
I also found a third-degree equivalent for Binet’s formula. There was certainly some patterns brewing here!
So I decided to consider the relationship between the sequence 1,1,2,4,8,15,29,56,… and the equation x⁴-x³-x²-x-1=0 and lo and behold, I found exactly the fourth-degree equivalents for the conjectures I obtained earlier, although it must be mentioned that proving these would be much more difficult than proving the previous conjectures.
I decided to write about my findings. These were not just mere mathematical explorations- they were mind-blowing ideas to my eyes, and a tribute in my heart to our mathematical ancestors who dealt with similar sequences and found ground-breaking theorems. If I could even 0.0000001% of that, I would consider myself grateful to my moments of sweet discovery.
My resulting piece of work was a paper on what I called m-bonacci sequences. There were few research papers on the same (I found one published in Ithaca College and even referenced it in my paper; in fact, some papers even called these n-bonacci), but I decided to publish my findings in a good Math journal so I could present my ideas on a well-respected and reputed platform.
My experience in trying to find a good journal for publishing my research was a roller-coaster ride in itself, but I leave that narration for some other day.
I request all my readers to read my paper, for reading and attempting to understand them would mean a good deal of appreciation for my work, irrespective whether somebody disputes my methods or not.
You can find the journal issue in Volume 6 Issue 3 of International Journal of Mathematics And its Applications. And the link to my paper: http://ijmaa.in/v6n3/357-363.pdf
I would appreciate constructive feedback to my research.
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