A brief introduction to the Fibonacci numbers.

Origins of the numbers

Definitions: the Fibonacci numbers can be rather simply defined by the following:

1. Start1 with the numbers 1 and  2.

2. Add them together to make the next number (3).

3. Then form the next number in sequence by adding the previous two together.

More formally: [F1=1; F2=2], [F3=3], [Fn=Fn-1+Fn-2].

The resulting sequence begins like this:

F1 to F19 :
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765...
(The boldface ones are prime. )

The numbers seem to pop up in perhaps unexpected places (e.g., cornfields,  seashells, Greek monuments).

A variety of unsolved problems concerning the numbers remain (e.g., "are there infinitely many prime Fibonacci's?", "what happens when the start numbers are varied?", "what sorts of sequences arise modulo n?").

1 It is interesting to note that while most modern presentations begin the sequence (1,1,2,3,...), Fibonacci himself omitted the initial 1 and began, instead, with F2=2.

to outline