Proving the infinitude of primes using
topological spaces
Ahd Al-Namer
1.Abstract
In this research I introduced a new way of proving the infinitude of
prime numbers by defining a topological space on the set of integers
and try to reach a point that will tell us clearly that the set primes
are infinitely many.
2.Introduction
In the period of 200-300 B.C Euclid introduced a proof of the
infinitude of prime numbers using the contradiction method, this
proof was mainly based on Set theory and Number theory.
But now the proof is about to be given to you is based on topology.
To make this publishment easier for everyone to understand I will
define what a topology is, what is an open set and closed set and
then I will define a certain topology and use it to reach the required.
3.Definitions
Definition 1:
Let 𝑋 be any non-empty set, 𝑈 ⊂ 𝑋, we say that (𝑋, 𝒯 ) is a
topological space if and only if :
1. 𝜙 , 𝑋 ∈ 𝒯
2. Any finite intersection of members of 𝒯 is also in 𝒯, i.e.
If 𝑈1 , 𝑈2 , … , 𝑈𝑛 ∈ 𝒯 , then ⋂𝑛𝑎=1 𝑈𝑎 ∈ 𝒯.
3. Let {𝑈𝛼 ∶ 𝛼 ∈ Δ} be a family of sets that each one is in 𝒯,then
the union of this family of sets is also in 𝒯.
Definition 2: Let (𝑋, 𝒯 ) be a topological space;
1. a set U is said to be open set if it is a member of (𝑋, 𝒯 ).
2. Set V is said to be a closed set if and only if 𝑋 − 𝑉 is open.
Now we are ready to define a topology to help us with our proof,
this topology is called the profinite topology or evenly-spaced
integer topology for some references.
Let (ℤ, 𝒯 ) be a set of subsets of ℤ including 𝜙, 𝑈 ⊂ ℤ : if a ∈ 𝑈 ,
then a+mℤ ⊂ 𝑈 m ≥ 1, i.e. if an element a is in U then it’s
equivalent class mod m is a subset of U.
Firstly, let’s show that this is a topological space:
1- 𝜙 ∈ 𝒯 , ℤ ∈ 𝒯 𝑠𝑖𝑛𝑐𝑒 ℤ = 0 + 1. ℤ .
2- Let 𝑈1 , 𝑈2 be open sets in 𝒯, let a ∈ 𝑈1 ∩ 𝑈2 since a ∈ 𝑈1 , 𝑈2 then
a+𝑚1 . ℤ ⊂ 𝑈1 and a+𝑚2 . ℤ ⊂ 𝑈2 .Now, a+𝑚1 . 𝑚2 ⊂ (a + 𝑚1 . ℤ ∧ a +
𝑚2 . ℤ) then a+𝑚1 . 𝑚2 ℤ ⊂ 𝑈1 ∩ 𝑈2 then 𝑈1 ∩ 𝑈2 ∈ 𝒯.
3- Let {𝑈𝛼 ∶ 𝛼 ∈ Δ } be a family of open sets. We need to prove that
⋃ 𝑈𝛼 ∈ 𝒯
𝛼∈Δ
Let a ∈ ⋃𝛼∈Δ 𝑈𝛼 ∈ 𝒯 , then a+𝑚𝛼 ℤ is a subset of the union since
for each 𝛼, 𝑈𝛼 is an open set, then:
⋃ 𝑈𝛼 ∈ 𝒯
𝛼∈Δ
Remark:
1-each open set in this topology is an infinite set
2-each set with the form of a+mℤ in this topology is both open and
closed set (Clopen set).
4.Proof
Now we are ready to give a proof for the infinitude of prime
numbers.
Consider the profinite topology on the set of integers, and consider
the following union:
⋃ 𝑝ℤ
𝑝
This union is the set of all multiples of primes, clearly ⋃𝑝 𝑝ℤ = ℤ −
{1, −1}; since {1,-1} are not a prime numbers nor a multiple of a
prime.
{1,-1} is not an open set thus ⋃𝑝 𝑝ℤ is a set of form ⋃𝑖 0 + 𝑝𝑖 ℤ is
not closed.
However, for each 𝑝𝑖 , 0+𝑝𝑖 ℤ is a closed set, then for any finite union
it must be a closed set, this leads us to the fact that ⋃𝑝 𝑝ℤ is an
infinite union, for which primes are infinite.∎
5.References
1. https://en.wikipedia.org/wiki/Furstenberg’s proof of the infinitude of primes.
2. H. Furstenberg, “On the Infinitude of Primes,” Amer. Math. Monthly 62 (1955), 353.
3. ”.KEITH CONRAD, “THE “TOPOLOGICAL” PROOF OF THE INFINITUDE OF PRIMES”