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Fibonacci numbers and the golden section - Article Example

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Fibonacci numbers and the golden section

Recall that an integer is prime if it has no proper divisors. Some Fibonacci numbers are prime, for example 514229, but it is still unknown whether there exist infinitely many prime Fibonacci numbers. The problem of finding prime numbers with many digits is crucial for the
find a very large prime number, you are able to write a secret code that is reasonably safe (this principle is the basis of the Public Key Cryptography, nowadays used by banks and governments all over the world).
Suggested readings. We suggest, as a first reading, the following website: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html. It is well written, in elementary terms, contains a number of illustrations and it explains clearly some applications of Fibonacci numbers to natural sciences. It contains also several links to other websites on the same topic. We suggest to follow the link "Fibonacci numbers in nature": there, you will find applications to family trees of rabbits, cows, geometry, flowers, and vegetables! It is a short, fascinating walk in the real world seen through the mathematicians' eyes.
As a further, more technical reading, one can read the material contained in the website: http://en.wikipedia.org/wiki/Fibonacci_number. ...
It is well written, in elementary terms, contains a number of illustrations and it explains clearly some applications of Fibonacci numbers to natural sciences. It contains also several links to other websites on the same topic. We suggest to follow the link "Fibonacci numbers in nature": there, you will find applications to family trees of rabbits, cows, geometry, flowers, and vegetables! It is a short, fascinating walk in the real world seen through the mathematicians' eyes.
At the bottom of the page there are suggestions on the paths to follow to explore further the site.
As a further, more technical reading, one can read the material contained in the website: http://en.wikipedia.org/wiki/Fibonacci_number. Part of this site is probably too advanced for a non-specialist, but most of its content is certainly accessible.
These readings can be the opportunity to learn a little, but very useful, piece of mathematics: the difference equations. A difference equation is a function whose value at n is
defined linearly by the value at n-1 and n-2, as in the case of Fibonacci numbers. For such functions, there exist always a closed formula, that is, a formula giving the value at n only as a function of n, with no knowledge of the values at n-1 and n-2. The method is explained at the beginning of http://en.wikipedia.org/wiki/Recurrence_relation.
In the bibliography, we suggest some elementary books for a further reading.
To conclude, I think that this suggested reading is accessible to everybody, it doesn't require any special knowledge in mathematics and it has sufficiently many practical applications in arts and science, to be a fascinating and intriguing subject.







BIBLIOGRAPHY

[1] Dunlap, R. The Golden Ratio and ...Show more

Summary

Why are these numbers interesting First of all, they show up in many practical problems, especially in biological contexts, such as the branching in trees, genealogy trees and the production and shape of fruits. But they are also related to the so-called 'golden section': given a segment [a,b], to find its golden section means to find a number x on that segment such that
Author : fred32
Fibonacci numbers and the golden section essay example
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