The third prime number, which is also the second Fermat prime, the third Sophie
Germain prime, and Fibonacci number 
. It is an Eisenstein
prime, but not a Gaussian prime, since it factors
as 
. It is the hypotenuse
of the smallest Pythagorean triple: 3, 4, 5.
For the Pythagorean school, the number 5 was the number of marriage, since it is
was the sum of the first female number (2) and the first male
number (3). The magic symbol of the pentagram
was also based on number 5; it is a star polygon
with the smallest possible number of sides, and is formed by the diagonals of a regular
pentagon. These intersect each other according to the golden
ratio 
.
There are five Platonic solids. In algebra, five arises in Abel's impossibility theorem
as the smallest degree for which an algebraic equation with general coefficients
is not solvable by radicals. According to Galois theory,
this property is a consequence of the fact that 5 is the smallest positive integer
such that the symmetric
group 
is not a solvable
group. Five is also the largest positive integer 
such that every finite group
of order 
is Abelian.
According to Weyl (1952; Chandrasekharan 1986) the five-fold symmetry is typical of plants and animals, whereas it does not appear in the inanimate world.
Words referring to number five often start with the prefix penta- (in Greek 
-), whereas terms like
quintic and quintuple are derived from the Latin quintus
(fifth).
