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Number Theory

The Fibonacci Numbers

Number theory deals with the analysis of integers, i.e. whole numbers without fractions or decimals. Many applications of modeling are by nature only valid for whole numbers and can never be parts of a number. For example, when counting people, or animals. They are ordered on the numbering line ranging from negative infinity, ... -3, -2, -1, 0, 1, 2, … up to positive infinity. 

One important application of number theory is in the Fibonacci Numbers, presented originally in the 13th century in Liber Abaci [1]. This is widely regarded as a catalyst in bringing a more modern understanding of numbers, introducing the Arabic numbering system to Europe [2]. The system was formulated by Italian mathematics professor Leonardo Pisano Fibonacci, a mathematician from the Italian city of Pisa and widely renowned as one of the greatest mathematicians of the Medieval era [3] [4].

Fibonacci studied rabbit populations and looked at a simplified model of the reproduction formulated as: [5]

  1. Rabbits grow and can produce offspring after one month;
  2. Each female produces two new offspring (one male, one female) each month;
  3. No rabbits ever die.

This lead to the world famous equation: [6]

This may seem rather moderate at first sight, but leads to very high numbers as the index n grows:

n

Fn

10

55

30

832040

50

12586269025

100

354224848179261915075

300

222232244629420445529739893461909967206666939096499764990979600

Table 1. Some Fibonacci Numbers [7]

Although the Fibonacci numbers were formulated on a very idealized and simple model, they have relevance in many forms of combinatorics problems including stock trading, art, computer science and biology.  

References

1. Fibonacci, L.P.: Liber Abaci. 1202.

2. The Life and Numbers of Fibonacci. Plus Maths, https://plus.maths.org/content/life-and-numbersfibonacci

3. Who was Fibonacci? Ron Knott, Surrey University, UK. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html

4. Devlin, K.: The Man of Numbers, Fibonaccis Arithmetic Revolution, Walker and Company 2011. 

5. Fibonacci and the rabbits the story. The Fibonacci Project, University of Bayreuth, Germany, http://fibonacci.uni-de/project/fibonacci-and-the-rabbits/the-story.html

6. Fibonacci Number, Wolfram Mathworld, http://mathworld.wolfram.com/FibonacciNumber.html

7. The First 2000 Fibonacci Numbers. The Online Encyclopedia of Integer Sequences, https://oeis.org/A000045/b000045.txt

 

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