Leonardo of Pisa was the Italian mathematician of the late twelfth century and early thirteenth century known famously as Fibonacci. In 1202, he published the book Liber Abaci in which he subtly presented the Fibonacci sequence through a problem regarding the reproduction of a one pair of rabbits. The problem stated: “A pair of adult rabbits produces a pair of baby rabbits once each month. Each pair of baby rabbits requires one month to grow to be adults and subsequently produces one pair of baby rabbits each month thereafter. Determine the number of pairs of adult and baby rabbits after some number of months. It is also assumed that the rabbits are immortal.” Edouard Lucas brought attention to the sequence again and called it the Fibonacci sequence in the nineteenth century. Lucas studied the sequence extensively and generalized to a case where $L_1 = 1, L_2 = 3, L_n = L_{n-1} + L_{n-2}$. The sequence is now called the Lucas sequence.
Although the sequence is named after him, Fibonacci was not the first one to have described it. Notions of the sequence appeared in India as early as 200 BC, in particular, with Sanskrit prosody.
By the twentieth century, interest in Fibonacci numbers rose in the mathematics community. The Fibonacci Association was founded in 1963 by mathematician Verner E. Hoggatt; the association began to publish the journal The Fibonacci Quarterly. International conferences to discuss Fibonacci numbers began in the 1980s.
We define the Fibonacci sequence $\{f_n\}$ as follows:
$$f_1 = f_2 = 1$$
$$f_{n+1} = f_n + f_{n-1}$$ where $n = 2, 3, ...$.
Fibonacci numbers have appeared throughout mathematics. They are used in looking at how long the Euclidian algorithm takes to run. Approximate conversations between miles and kilometers can be made if the miles given happens to be a Fibonacci number. In such a case, it’s conversion to kilometers is approximately the next Fibonacci number in the sequence. The sequence can be found by adding the entries along each shallow diagonal of Pascal’s triangle. Variations of the Fibonacci sequence include the Negafibonacci numbers (indexing includes negative integers), the Lucas numbers mentioned earlier, and the Pell numbers ($p_n = 2p_{n-1} + p_{n-2}$).
Apart from their many applications within mathematics, Fibonacci numbers appear vastly throughout nature. If we were to look at the arrangement of petals in flowers, they come in groups of 3, 5, 8, 13, $\dots$. These are remarkable the Fibonacci numbers. Having flower follow this patter allows for optimization of space given the time seeds will appear in the flower head’s center. Leaves on the stems of plants are also arranged in a similar way.
The sequence is also connected to the golden ratio $\phi = \frac{1+ \sqrt{4}}{2}$. In particular, if we form a sequence using ratios from terms of the Fibonacci sequence, this sequence $\{\frac{f_{n+1}}{f_n}\}$ approaches the golden ratio as $n$ approached infinity.
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Right now I've in looking at generalized Fibonacci numbers, which are defined in a couple of the papers in the Dropbox. I would really like to incorporate field/Galois theory into my research while looking at this, so when I search for articles right now this is what I'm looking for. I don't have a specific research question yet.
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