Furstenberg's proof of the infinitude of primes

Define a topology on the integers $\backslash mathbb\{Z\}$, called the evenly spaced integer topology, by declaring a subset U ⊆ $\backslash mathbb\{Z\}$ to be an open set if and only if it is a union of arithmetic sequences S(a, b) for a ≠ 0, or is empty (which can be seen as a nullary union (empty union) of arithmetic sequences), where

:$S(a,\; b)\; =\; \backslash \{\; a\; n\; +\; b\; \backslash mid\; n\; \backslash in\; \backslash mathbb\{Z\}\; \backslash \}\; =\; a\; \backslash mathbb\{Z\}\; +\; b.$

Equivalently, U is open if and only if for every x in U there is some non-zero integer a such that S(a, x) ⊆ U. The axioms for a topology are easily verified:

• ∅ is open by definition, and $\backslash mathbb\{Z\}$ is just the sequence S(1, 0), and so is open as well.
• Any union of open sets is open: for any collection of open sets U_i and x in their union U, any of the numbers a_i for which S(a_i, x) ⊆ U_i also shows that S(a_i, x) ⊆ U.
• The intersection of two (and hence finitely many) open sets is open: let U₁ and U₂ be open sets and let x ∈ U₁ ∩ U₂ (with numbers a₁ and a₂ establishing membership). Set a to be the least common multiple of a₁ and a₂. Then S(a, x) ⊆ S(a_i, x) ⊆ U_i.

This topology has two notable properties:

1. Since any non-empty open set contains an infinite sequence, a finite non-empty set cannot be open; put another way, the complement of a finite non-empty set cannot be a closed set.
2. The basis sets S(a, b) are both open and closed: they are open by definition, and we can write S(a, b) as the complement of an open set as follows:

::: $S(a,\; b)\; =\; \backslash mathbb\{Z\}\; \backslash setminus\; \backslash bigcup\_\{j\; =\; 1\}^\{a\; -\; 1\}\; S(a,\; b\; +\; j).$

The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.

::: $\backslash mathbb\{Z\}\; \backslash setminus\; \backslash \{\; -1,\; +\; 1\; \backslash \}\; =\; \backslash bigcup\_\{p\; \backslash mathrm\{\backslash ,\; prime\}\}\; S(p,\; 0).$

Now, by the first topological property, the set on the left-hand side cannot be closed. On the other hand, by the second topological property, the sets S(p, 0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a contradiction, so there must be infinitely many prime numbers.

The evenly spaced integer topology on $\backslash Z$ is the topology induced by the inclusion $\backslash Z\backslash subset\; \backslash hat\backslash Z$, where $\backslash hat\backslash Z$ is the profinite integer ring with its profinite topology.

It is homeomorphic to the rational numbers $\backslash mathbb\{Q\}$ with the subspace topology inherited from the real line,{{Cite journal |last=Broughan |first=Kevin A. |date=August 2003 |title=Adic Topologies for the Rational Integers |journal=Canadian Journal of Mathematics |language=en |volume=55 |issue=4 |pages=711–723 |doi=10.4153/CJM-2003-030-3 |issn=0008-414X |s2cid=121286344|doi-access=free }} which makes it clear that any finite subset of it, such as $\backslash \{-1,\; +1\backslash \}$, cannot be open.

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| last1 = Mercer | first1 = Idris D.

| title = On Furstenberg's Proof of the Infinitude of Primes

| journal = American Mathematical Monthly

| volume = 116

| issue = 4

| pages = 355–356

| year = 2009

| doi = 10.4169/193009709X470218

| url = http://www.idmercer.com/monthly355-356-mercer.pdf

| citeseerx = 10.1.1.559.9528

}}

}}

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| first = Harry

| authorlink = Hillel Furstenberg

| title = On the infinitude of primes

| journal = American Mathematical Monthly

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| jstor = 2307043

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• {{Cite journal

| last1 = Mercer | first1 = Idris D.

| title = On Furstenberg's Proof of the Infinitude of Primes

| journal = American Mathematical Monthly

| volume = 116

| issue = 4

| pages = 355–356

| year = 2009

| doi = 10.4169/193009709X470218

| url = http://www.idmercer.com/monthly355-356-mercer.pdf

| citeseerx = 10.1.1.559.9528

}}

• Keith Conrad https://kconrad.math.uconn.edu/blurbs/ugradnumthy/primestopology.pdf
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• [http://www.everything2.com/index.pl?node_id=1460203 Furstenberg's proof that there are infinitely many prime numbers] at Everything2
• {{PlanetMath|urlname=FurstenbergsProofOfTheInfinitudeOfPrimes|title=Fürstenberg's proof of the infinitude of primes}}

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