# Fibonacci Numbers (and the Euler-Binet Formula) [Video]

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Dave4Math » Number Theory » Fibonacci Numbers (and the Euler-Binet Formula) [Video]

This article introduces the Fibonacci numbers and shows how to use mathematical induction to prove several Fibonacci identities. I then develop the Euler-Binet Formula involving the golden-ratio. Afterward, I discuss the Fibonacci Prime Conjecture and the growth of the Fibonacci sequence.

In this video, I explain what the Fibonacci numbers are and how they arise. Because the Fibonacci sequence have many exciting properties, I demonstrate several of them in this video, many of them using mathematical induction. Then I explain the golden ratio and its’ connection with the Fibonacci sequence. After that, I state and prove the Euler-Binet Formula. Finally, I explore the Fibonacci Prime Conjecture and the growth of the Fibonacci sequence.

Hi everyone, welcome back to my channel, I’m Dave.

In today’s episode, I define the Fibonacci numbers as a recursive sequence starting with zero and one and then adding the previous two integers together. Because of the beauty, many people have long noticed that the Fibonacci sequence arises in many places throughout the natural world. In other words, the Fibonacci sequence has many unique mathematical properties.

Before you continue, you may want to checkout the video notes here on Fibonacci Numbers. You may also be interested to know that this video is part of the Number Theory Series playlist.

## Fibonacci Numbers (and the Euler-Binet Formula)

### How-to Learn Fibonacci Numbers

By studying this video, you learn about the Fibonacci numbers and many of its properties.

### Step 1: Learn what the Fibonacci Numbers are.

Their definition is recursive, so make sure and work out some basic examples.

### Step 2: Practice proving some Fibonacci identities using mathematical induction.

So, here are some examples of this.

### Step 3: Explore the the Euler-Binet formula.

After that, learn that the golden ratio is the positive root of a quadratic equation and that the Euler-Binet formula gives a relationship between the golden ratio and Fibonacci numbers.

### Step 4: Understand that there are many open questions about the Fibonacci sequence.

After that, you should realize that there are many open questions involving the Fibonacci sequence. In particular, you will learn about the Fibonacci sequence’s growth, particularly about exponential growth and a fundamental inequality between the n-th Fibonacci number and the n-th power of the golden ratio.

## What are the Fibonacci numbers?

The Fibonacci numbers are recursively defined. Starting at 0 and 1, and then from here, add the previous two Fibonacci numbers to get the next Fibonacci number. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

## Why is the Fibonacci sequence so important?

Many natural things follow the Fibonacci sequence, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, and an artichoke flowering. In some hurricanes and galaxies, the body rotation spawns spiral shapes (that follow the Fibonacci sequence) when the center turns faster than the periphery.

## How many Fibonacci prime numbers are there?

The conjecture is that there is an infinite number of Fibonacci numbers that are prime numbers, such as 19, 134, 702, 400, 093, 278, 081, 449, 423, 917. But no one knows for sure. This question is still an open conjecture.

## What is the Euler-Binet formula?

The Euler-Binet formula is a closed formula for finding the n-th Fibonacci number (defined recursively). The formula states that the n-th Fibonacci number is equal to the difference of the n-th powers of the golden ratio and its conjugate divided by the square root of 5.

In conclusion, I want to turn it over to you.

Do you think there are infinitely many Fibonacci primes?

So, either way, let us know what you think in the comments for this video: Fibonacci Numbers right now.

In conclusion, I want to turn it over to you.

Do you think there are infinitely many Fibonacci primes?

So, either way, let us know what you think in the comments for this video: Fibonacci Numbers right now.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.