In this post we discuss a proof for the infinitude of primes that uses topology (due to Furstenberg, see [2]). This is an interesting proof that gives a topological view point to a very familiar and basic mathematical fact. It is an elegant proof that is worthy to be considered as straight from “The Book” by Paul Erdos (at least “The Approximate Book”). Specifically it is one of the six proofs for the infinitude of primes found in [1].
Let be the set of all integers and
be the set of all prime numbers. The key to the proof is to define a topological space on
such that the assumption that
is finite will contradict some fact about this topological space.
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Defining the Topological Space
For any with
, define
. Let
be the set of all possible
. We show that
is a base for a topology on the space
.
We can visualize by thinking
being in the center and the other points of
are generated by adding integer multiples of
to the center
(in both the positive and negative directions). A useful observation is that we can make any point in
the center and we can still generate the same set
. Specifically, for each
, we have
.
We now show that is a base for a topology on
. Clearly
is a cover of
. Next we show that if
, then there is some
with
.
Let . Based on the observation made above, we have
and
. So we have
. Observe that
.
Let denote the space
with the topology generated by the base
. We need the following facts:
- Every non-empty open subset of
is infinite.
- Every
is a closed set in
The first bullet point is clear since every non-empty open set would have to contain one , which is infinite. To see that
is closed, let
. Observe that:
Thus the complement of is open is
.
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The Infinitude of Primes
We now tie this topological space to the prime numbers. Recall that denotes the set of all prime numbers. Also recall that any integer not equaled to 1 or -1 has a prime divisor. Thus we have the following:
If is finite, then the left-hand side of the above equation is a closed set (being the union of finitely many closed sets). This implies that
is an open set in the space
, which is impossible since every non-empty open set in this space is infinite. Thus
must be infinite.
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Some Comments About This Topological Space
The topological space generated by the base
has nice topological properties. It is Hausdorff. It is regular since the base
consists of open and closed sets. It is a separable metric space since it has a countable base. The space
is non-discrete at every point. Thus it differs from the usual topology on the integers.
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Reference
- Aigner, M., Gunter, M.Z. Proofs from THE BOOK, third edition, Springer-Verlag, Berlin, 2004.
- Furstenberg, H., On the infinitude of primes, Amer. Math. Monthly, 62, 353, 1955.
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