Leonardo de Pisa (1180-1250?), better known as Fibonacci, wrote the Liber Abaci, in which he included a problem about rabbits:
A man put one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year, if the nature of these rabbits is such that every month each pair bears a new pair which from the second month on becomes productive?
Ignoring the terrible incestuous implications, the resulting sequence of numbers of pairs of rabbits is known as the Fibonacci numbers:
This works out to the recursive sequence
for 
, where 
, the first known recursive definition in
mathematics.
An important result which we will need in the following theorems is this:
Proof: by induction on n.
Theorem 13.1: For the Fibonacci sequence,
for every 
.
Proof: direct, and using lemma, p. 27.
Theorem 13.2: For 
and , 
.
Proof: by induction on n (straightforward, using (1)).
Lemma: If m = qn+r, then
Theorem 13.3: The greatest common divisor of two Fibonacci numbers is
again a Fibonacci number; specifically
where
.
Corollary: In the Fibonacci sequence, 
if and only if m | n
for 
.
- For every prime p, there are infinitely many Fibonacci numbers that are divisible by p, equally spaced along the Fibonacci sequence.
-
It is not known if there are infinitely many prime Fibonacci numbers.