1, 1, 2, 3, 5, 8, 13, … This sequence may seem familiar to us all. It is the Fibonacci sequence. Today, we explore 5 intriguing facts about Fibonacci numbers in this article.
Table of Contens
- How the Fibonacci Sequence goes on
- The golden ratio
- The factors of Fibonacci numbers
- Fibonacci primes
- Perfect powers that are Fibonacci numbers
1. How the Fibonacci Sequence Goes on
Despite how much the Fibonacci sequence displays mathematical beauty, it goes on in a surprisingly easy-to-understand way. Start with the numbers 1 and 1, respectively. Add them together, and you get 2. Add 1 (the penultimate term of the sequence) and 2 (the last term of the sequence) together, and you get the next one, which is 3. This goes on forever and forever. The two 1’s in the sequence are defined as the first and second term of the sequence, while 2 is the third and 3 is the fourth, etc. Specifically, the first few terms of the sequence are listed below.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
2. The Golden Ratio
If you repeatedly divide the (n+1)th term of the Fibonacci sequence by the nth term of this sequence, you might notice that it gets closer to a certain number. That number is a golden ratio, and it’s irrational, meaning that it is not expressible as fractions of integers. And this number that looks arbitrary is no typical irrational number.
The golden ratio can be expressed as:
If you subtract 2 from the golden ratio, it is expressible as:
These two expressions both involve irrational numbers (sqrt(5)), yet when they are compared, you can see something so similar in them. That is the beauty of the golden ratio, which makes the famous spiral formed when connecting Fibonacci numbers.
3. The Factors of Fibonacci Numbers
The parity of the Fibonacci sequence is always odd, odd, even, odd, odd, even, and so on. But why does this pattern exist? That’s because of the remarkable similarities between the sequence of numbers and the sequence of Fibonacci numbers in terms of the relationships of their factors.
Suppose the set of factors of n be m. In that case, Fn is divisible by all Fibonacci numbers whose index is in m. If you find it hard to understand, here are a few examples. For instance, F10 = 55, and it’s divisible by F5 = 5, F2 = 1, and F1 = 1. Additionally, no two consecutive Fibonacci numbers have a gcd of greater than 1, which is another reason this sequence is similar to the sequence of integers in its factors.
4. Fibonacci Primes
Because if a Fibonacci number is prime, its index must also be prime unless it is 4, it’s pretty natural to think that a decent amount of primes exist that are also Fibonacci numbers. However, these primes are surprisingly rare. In fact, of the billions and billions of prime numbers that exist in the range where they are enumerated completely, only 51 Fibonacci numbers are known to be prime. It has not even been proven that there exist infinitely many Fibonacci primes, even if we have multiple ways to show that there are infinitely many primes.
5. Perfect Powers that are Fibonacci Numbers
For sequences that do not directly relate to perfect powers like the Mersenne numbers, there are surprisingly few perfect powers among Fibonacci numbers. In fact, it has been said to be proven that the only perfect powers in the infinite set of Fibonacci numbers are 1, 8, and 144. That’s it. For some reason, the Fibonacci numbers avoid perfect powers like in the case of Fibonacci primes. It is also notable that the indices of two of the numbers are their square roots, namely F1 = 1 and F12 = 144.
Conclusion
In this article, we’ve explained 5 intriguing facts about the Fibonacci sequence, which is a sequence that is very commonly mentioned and even seen in nature. Thus, this sequence exhibits so-called mathematical beauty.
References and Credits
- (n.d.). Fibonacci Sequence. Retrieved February 24, 2022, from https://www.mathsisfun.com/numbers/fibonacci-sequence.html
- (2006, December 6). Fibonacci numbers at most one away from a perfect power. Retrieved February 24, 2022, from https://homepages.warwick.ac.uk/~maseap/papers/fnpm.pdf