Maybe the roughest lesson, someone can ever receive, is this:

The implications related to that simple line, can fill a whole library: One of them, is the, so called, Fibonacci sequence:

Defined as:

Tabulating the ratio between two consecutive terms of the sequence, appear on insight, two properties:

1) This ratio has a finite limit.

2) Two consecutive Fibonacci numbers are

relatively prime.

To prove the first property, we have:

With positive solution, equal to the golden ratio.

The proof, of this second property, used to be by induction or contradiction [2], but this can also be proved using the oldest procedure designed to calculate de greatest common divisor, GCD, of two integers, known as **The Euclidean Algorithm**

The Euclidean method constructs a decreasing sequence of integers who share the same GCD.

if then

Where the function is the **Floor Function**, see [3].

*Proof:*

The Euclidean Algorithm reproduces all FibonacciÂ´s sequence and proves that two consecutive terms are relatively prime.

**References:**
[1]-Fundamentals of Number Theory William J. LeVeque – Courier Dover Publications, 1996 – ISBN 0486689069, 9780486689067

[2]-Consecutive Fibonacci Numbers Relatively Prime – The Math Forum@Drexel

[3] Weisstein, Eric W. “Floor Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FloorFunction.html

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