Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?

Perhaps it's not an entirely practical application, but Fibonacci numbers can be used to convert from miles to kilometers and vice versa:

Take two consecutive Fibonacci numbers, for example 5 and 8. And you're done converting. No kidding – there are 8 kilometers in 5 miles. To convert back just read the result from the other end - there are 5 miles in 8 km!

But why does it work?

Fibonacci numbers have a property that the ratio of two consecutive numbers tends to the Golden ratio as numbers get bigger and bigger. The Golden ratio is a number and it happens to be approximately 1.618.

Coincidentally, there are 1.609 kilometers in a mile, which is within 0.5% of the Golden ratio.

Consecutive Fibonacci numbers are the worst-case (maximum number of steps) numbers for Euclid's gcd algorithm.

Suppose you're writing a computer program to search a sorted array for a particular value. Usually the best method to use is a binary search. But binary search assumes it's the same cost to read from anywhere in the array. If it costs something to move from one array element to another, and the cost is proportional to how many array elements you need to skip over to get from one element you read to the next, then Fibonacci search works better. This can apply to situations like searching through arrays that don't fit entirely in your computer's cache so it's generally cheaper to read nearby elements that distant ones.

Yep, the solution by Matiyasevich to Hilbert's tenth problem relies heavily on showing exponentiation is Diophantine. In other words, the relation R given by $R(x,y,z)$ if and only if $x^y = z$ can be expressed by a polynomial equation $T(x,y,z,w)=0$, such that $R(x,y,z)$ holds if and only if $T(x,y,z,w)=0$ has a solution with $w$ integral. This is a very difficult and technical result, as well as rather counterintuitive.

Recall that the Fibonacci numbers grow essentially exponentially. One of the lemmas in showing that exponentiation is Diophantine is to show that the Fibonacci sequence (and its variants) are Diophantine. See the book by Matiyasevich on Hilbert's 10th problem.