## Saturday, March 3, 2012

### Fibonacci Day: Combining Two Famous Sequences into One Cool Math Stuff Post

Finally, we have reached another Fibonacci day. It is March 3, and 3 is the fourth Fibonacci number. Let’s look at the Fibonacci numbers again, as they may have faded from our memories over the past month.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,...
There is also another series of numbers that we did look at back in August called the prime numbers. If you don’t remember them, they are natural numbers that are only divisible by one and itself. For a quick review, 5 is prime because it is only divisible by 1 and 5. 6 is not prime because it is divisible by 1, 2, 3, and 6.
Here are the prime numbers:
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41,...
These are two famous infinite series that are both very significant in science and mathematics. Let’s see what numbers are in both series.
2, 3, 5, 13, 89, 233, 1597, 28657, 514229,...
That’s interesting. There is one very cool thing about these numbers though, that is kind of hard to find at first glance. It isn’t about the numbers themselves, but their position in the Fibonacci sequence. Here are their positions:
3, 4, 5, 7, 11, 13, 17, 23, 29
Do you see a pattern? They are all prime numbers, except for that 4 at the beginning. The four is the only exception as far as I know to this rule. If you notice another exception, then please comment and let us know.
If you look at it vice versa though, you don’t see this beautiful pattern though. If you look at the second Fibonacci number, we have 1, which is not prime. It is actually called a universal number.
Also, the nineteenth Fibonacci number isn’t prime. While 4181 looks very prime, it is actually 113 x 37.
There are a bunch more of these, which I will list below:
31. 1346269 = 557 x 2417
37. 24157817 = 10877 x 2221
41. 165580141 = 2789 x 59369
53. 53316291173 = 953 x 55945741
Though there are many exceptions, I find it pretty cool that almost every prime Fibonacci number is in a prime position.
Bonus: Here is a challenge for you. We know that there is an infinite quantity of Fibonacci numbers. In August, we proved there is an infinite quantity of prime numbers. But are there infinitely many prime Fibonacci numbers?
This is a question that mathematicians still have not figured out. If you were to figure this out, you would have eternal fame in the world of mathematics. I’d say that’s a pretty interesting fun fact. (You don’t have to do the challenge)