604 Infinitude of prime numbers
21.3 F ̈urstenberg’s topological proof made easy .......
There is a famous proof of the infinitude of primes using topology. It
can be found in many books. Apart from an introductory sentence, here
is the entire article of [F ̈urstenberg]:
We introduce a topology into the space of integersS, by using the arithmetic pro-
gressions (from−∞to+∞) as a basis. It is not difficult to verify that this actually
yields a topological space. In fact, under this topology,Smay be shown to be nor-
mal and hence metrizable. Each arithmetic progression is closed as well as open,
since its complement is the union of other arithmetic progressions (having the same
difference). As a result, the union of any finite number of arithmetic progression
is closed. Consider now the setA=
S
Ap, whereApconsists of all multiples of
p, andpruns through the set of primes≥ 2. The only numbers not belonging to
Aare− 1 and 1, and since the set{− 1 , 1 }is clearly not an open set,Acannot be
closed. HenceAis not a finite union of closed sets which proves that there are an
infinity of primes.
The most recent issue of Mathematics Magazine^2 contains a para-
phrase of this proof, avoiding the language of topology.
LetZbe the set of integers. We say that a subsetA⊂Zis periodic
if there is an integerksuch that for every integern∈Z,n∈Aif and
only ifn+k ∈ A. In other words, a period set is a (finite) union of
doubly infinite arithmetic progressions of the same common difference.
The following are clear.
1.IfAis periodic, then so is its complementZ\A.2.IfAandBare periodic sets, so is their union, the period of the
union being the lcm of the periods of the sets. For example, if
A 3 :={ 3 n:n∈Z}andA 5 :={ 5 n:n∈Z}, thenA 3 ∪A 5 is
the union of the arithmetic progressions of common difference 15,
containing the terms 0, 3, 5, 6, 9, 10.3.(2) extends to finite unions.For each prime numberp, letAp :={np:n∈Z}consists of the
multiples ofp. This is clearly periodic. Suppose there are only finitely
many primes. Then, the (finite) unionA:=
⋃
pprimeApis periodic, and
so is its complement. This complement is clearly the set{− 1 , 1 }, which,
being finite, cannot be periodic. This contradiction shows that there are
indeed infinitely many primes.
(^2) Math. Mag., 76 (2003) number 3.