Fibonacci Number Sequence and Prime Numbers

Please don’t let this title turn you away from the post.  Its a really great post, I promise . . . I just couldn’t come up with something cute and catchy today!

Anyway, I’ve been very vocal about the fact that NCTM asks great questions on their Facebook and Twitter pages, but today I have another Tweet that I really enjoyed from Maths Jam:

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So, this is a great question, right?  You probably know what makes a number a perfect square (just in case you don’t, look here), but you might not know about the Fibonacci Number Sequence.  I sort of wish that we’d talked about it before today, because there are many great and interesting things about this sequence of numbers, but we’ll just have to talk in more detail about those great and interesting things later.  For the purposes of this question you just need to know that the Fibonacci number sequence is a sequence of numbers which is generated by adding the previous two numbers together.  So terms 1 – 5 of the Fibonacci Number sequence are:

1, 1, 2, 3, 5, . . .

It seems that this particular Tweet asserts that the number sequence is actually

0, 1, 1, 2, 3, 5, . . . which I’d never seen before, until I did a little investigating via Wolfram Mathworld, but it isn’t really central to the answering of this question.

So, just to recap; we’re looking for numbers in this sequence that are also perfect squares.  In that case, I guess we should start by listing the Fibonacci Numbers, starting with the “0” term:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, . . .

Here’s the thing about answering this question, the number sequence generates an infinite number of terms meaning that, yes–I would venture a guess that at some point there will be another perfect square number in this sequence of numbers, but I’m not sure what it is.

This person calculated and factored the first 300 Fibonacci Numbers; and from this list it looks like 0, 1, and 144 are the only Fibonacci Numbers that are also perfect squares.

There are proofs, involving Lucas Numbers that also show that the only Fibonacci Numbers that are also perfect squares are 0,1,144.  That means that with the use of mathematics we are able to prove things about numbers that we know exist, but that no one has discovered yet.  Wow.  Just wow.

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3 thoughts on “Fibonacci Number Sequence and Prime Numbers

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