Furstenberg’s Topological Proof of the Infinitude of Primes

One surprising proof of the infinitude of primes is Furstenberg’s topological proof. At first glance it looks like a clever trick: put a strange topology on the integers, observe that arithmetic progressions are open and closed, and then use the fact that a finite union of closed sets is closed. But the proof is more than a trick. It is a small example of a very powerful way of thinking in modern mathematics: arithmetic information can often be encoded as topological information. Congruence conditions become neighborhoods. Divisibility conditions become closed sets. Finite arithmetic tests become local data. Infinite arithmetic phenomena become statements about closure, density, compactness, and limits.

The usual topology on the integers is inherited from the real line. In that topology, two integers are close if their ordinary distance is small. Thus $1000$ is far from $1$ , while $2$ is close to $1$ . But number theory often does not care about ordinary distance. It cares about congruences. From an arithmetic point of view, two integers may be very far apart in size and yet almost indistinguishable modulo many numbers. For example, $1+1000!$ is enormous, but modulo every integer $m\leq 1000$ , it behaves exactly like $1$ . Furstenberg’s topology makes this arithmetic idea precise.

Furstenberg’s topology

We define a topology on $\mathbb{Z}$ by taking as basic open sets the two-sided arithmetic progressions

$\displaystyle U(a,b)=a\mathbb{Z}+b=\{an+b:n\in\mathbb{Z} \}, \quad a\geq 1.$

In this topology, a neighborhood of an integer $b$ is not a small interval around $b$ . Instead, it is a congruence class containing $b$ . To say that $x$ is near $b$ means that $x$ satisfies the same congruence as $b$ modulo some integer $a$ :

$\displaystyle x\equiv b \pmod a.$

Thus the topology changes the meaning of closeness. It does not measure physical or geometric distance on the number line. It measures arithmetic indistinguishability under finite congruence tests. Two integers are close if, for a chosen modulus, they lie in the same residue class. Two integers are very close if they agree modulo many moduli, or modulo a very large modulus. The topology is not arbitrary. It is designed so that congruence classes become open sets. Since congruence classes are the basic objects of elementary number theory, the topology turns elementary arithmetic into geometry.

The family of sets $a\mathbb{Z}+b$ really does form a basis for a topology. The key point is the Chinese remainder theorem. If two congruence conditions are imposed, $x\equiv b \pmod a$ and $x\equiv d \pmod c,$ then either they are incompatible, or their simultaneous solutions form a single congruence class modulo ${\text{lcm}}(a,c)$ . In other words, the intersection of two basic open sets is either empty or again a basic open set of the same type. This is exactly the finite-intersection property needed for a basis.

There are two simple topological facts on which Furstenberg’s proof rests. First, every nonempty open set is infinite. Indeed, every nonempty open set contains some arithmetic progression $a\mathbb{Z}+b$ , and such a progression contains infinitely many integers in both directions. Therefore no nonempty finite subset of $\mathbb{Z}$ is open in this topology.

Second, every basic open set is also closed. This is because the complement of one residue class modulo $a$ is the union of the remaining residue classes modulo $a$ :

$\displaystyle \mathbb{Z}\setminus(a\mathbb{Z}+b)=\bigcup_{{r=0,\dots,a-1; ~ r\not\equiv b\pmod a}}(a\mathbb{Z}+r).$

So $a\mathbb{Z}+b$ is open by definition, and its complement is open as well. Hence $a\mathbb{Z}+b$ is closed. Such a set is called clopen: both closed and open.

This clopen property is extremely important. It means that congruence conditions are topologically very rigid. A condition such as $x\equiv b \pmod a$ defines a whole separated piece of the integer line. Once we look modulo $a$ , the integers split into finitely many disjoint clopen pieces:

$\displaystyle \mathbb{Z}=a\mathbb{Z}\cup(a\mathbb{Z}+1)\cup\cdots\cup(a\mathbb{Z}+a-1).$

So the topology sees the integers as being repeatedly cut into finite congruence patterns. The ordinary line is replaced by a space whose geometry is controlled by modular arithmetic.

Infinitude of Primes

Now we prove the infinitude of primes. Every integer except $1$ and $-1$ is divisible by some prime. Therefore

$\displaystyle \mathbb{Z}\setminus\{-1,1\}=\bigcup_{p\ \mathrm{prime}}p\mathbb{Z}.$

Each set $p\mathbb{Z}$ is a congruence class, hence closed. If there were only finitely many primes, say $p_1,\dots,p_k$ , then the union

$\displaystyle p_1\mathbb{Z}\cup\cdots\cup p_k\mathbb{Z}$

would be a finite union of closed sets, hence closed. Therefore its complement would be open. But its complement would be exactly the set of integers not divisible by any prime. If $p_1,\dots,p_k$ were all the primes, then that complement would be ${-1,1}.$ Thus ${-1,1}$ would be open. This is impossible, because no nonempty finite subset of $\mathbb{Z}$ is open in this topology. Therefore there must be infinitely many primes.

The proof is short, but it is significant. It says that a finite list of prime divisibility obstructions cannot account for all non-units of $\mathbb{Z}$ . Divisibility by a prime $p$ is the closed condition $x\in p\mathbb{Z}.$ The set of all non-units is the union of all these closed conditions. If there were only finitely many primes, then being a non-unit would be a finite closed congruence condition. Equivalently, being a unit would be an open congruence condition. But the units of $\mathbb{Z}$ are only $\pm 1$ , and a nonempty open condition must contain an entire arithmetic progression. The two-point set $\{-1,1\}$ is too small to be open. So this is the proof in one sentence: A finite set of congruence obstructions can never isolate only $\pm1$ .

This also explains the relation to Euclid’s classical proof. Euclid’s proof begins by assuming that there are finitely many primes, $p_1,\dots,p_k,$ and then considers

$\displaystyle N=p_1p_2\cdots p_k+1.$

The number $N$ is not divisible by any of the primes on the list. Therefore either $N$ itself is prime, or it has a prime divisor not on the list. In either case, the original list was incomplete. Furstenberg’s proof contains the same arithmetic mechanism, but it reveals a larger structure. Euclid produces one integer outside the finite list of prime divisibility classes. Furstenberg observes that there is not merely one such integer. There is a whole open arithmetic progression of them:

$\displaystyle 1+p_1p_2\cdots p_k\mathbb{Z}.$

Every integer in this progression is congruent to $1$ modulo each $p_i$ . Hence none of them is divisible by any $p_i$ . So the finite list of primes misses not just a single cleverly chosen number but an entire neighborhood in the arithmetic topology. This is the topological improvement of the Euclidean idea. Euclid says: from a finite list of primes, I can construct a number escaping the list. Furstenberg says: from a finite list of primes, I can construct an entire open set escaping the list. The contradiction arises because if the finite list were complete, the only escaping integers would be $\pm 1$ , but $\pm 1$ cannot form an open set.

The topology therefore turns Euclid’s argument into a statement about largeness. In this topology, a set is not large because it contains large numbers. A set is large because it contains a complete congruence class. Every nonempty open set is arithmetically large: it contains infinitely many integers arranged in a regular arithmetic pattern. Thus Furstenberg’s proof says that avoiding finitely many primes is always an arithmetically large condition. Only by avoiding all primes can one shrink down to the tiny set $\{-1,1\}$ .

Limits and Profinite Topology

The topology also gives a new way of thinking about limits. In the usual topology, the sequence $N!$ goes to infinity. But in the arithmetic topology it converges to $0$ . Indeed, fix any modulus $m$ . Once $N\geq m$ , the integer $m$ divides $N!$ , so $N!\equiv 0\pmod m.$ Therefore, no matter what congruence neighborhood of $0$ we choose, all sufficiently large factorials lie in that neighborhood. Similarly,

$\displaystyle 1+N!\to 1$

in this topology, because $1+N!\equiv 1\pmod m$ for all sufficiently large $N$ .

This is a very important idea. A limit does not always mean that numbers become close in size. A limit means that numbers eventually become indistinguishable by the tests defining the topology. In the usual topology, the tests are intervals. In the arithmetic topology, the tests are congruence classes. Thus the sequence $1+N!$ approaches $1$ because every fixed congruence test eventually sees it as equal to $1$ .

This leads naturally to the profinite completion of the integers. The topology generated by congruence classes is closely related to the profinite topology. The profinite completion of $\mathbb{Z}$ is

$\displaystyle \widehat{\mathbb{Z}}=\varprojlim_n \mathbb{Z}/n\mathbb{Z}.$

An element of $\widehat{\mathbb{Z}}$ is not necessarily an ordinary integer. It is a compatible system of residues modulo every positive integer $n$ . In other words, a profinite integer is a sequence

$\displaystyle (x_n)_{n\geq 1}, \qquad x_n\in\mathbb{Z}/n\mathbb{Z},$

such that these residues are compatible when one modulus divides another. Ordinary integers give such compatible systems: an integer $x$ determines the system

$\displaystyle (x\bmod n)_{n\geq 1}.$

But the completion also contains limiting objects that may not come from ordinary integers. This is analogous to how the real numbers complete the rational numbers with respect to the usual notion of distance. The real number line fills in the missing limits of Cauchy sequences of rationals. The profinite integers fill in the missing limits of integer sequences that become stable modulo every $n$ . The real numbers complete arithmetic with respect to size; the profinite integers complete arithmetic with respect to congruences.

An example of a limit that doesn’t correspond to an ordinary integer is $\displaystyle x=(0\in\mathbb Z_2,\ 1\in\mathbb Z_3,\ 1\in\mathbb Z_5,\ 1\in\mathbb Z_7,\dots).$ This cannot come from an ordinary integer. If an integer $m\in\mathbb Z$ satisfied $m\equiv 0\pmod {2^k}$ for every $k$ , then $m$ would have to be divisible by arbitrarily high powers of $2$ . The only ordinary integer with that property is $m=0.$ But $0$ does not satisfy $m\equiv 1\pmod 3.$ So this profinite integer is not an ordinary integer.

This is a powerful shift in perspective. In $\widehat{\mathbb{Z}}$ , every compatible system of finite congruence data has a limiting object. This is an infinite form of the Chinese remainder theorem. The ordinary Chinese remainder theorem says that finitely many compatible congruences can be solved modulo a common modulus. The profinite viewpoint says that infinitely many compatible congruence conditions determine a point in the profinite completion.

The prime-by-prime version of this idea gives the $p$ -adic integers. For a fixed prime $p$ , the $p$ -adic topology says that two integers are close if they are congruent modulo a high power of $p$ . Thus $x$ and $y$ are very close if

$\displaystyle x-y\equiv 0\pmod {p^k}$

for large $k$ . The $p$ -adic integers are

$\displaystyle \mathbb{Z}_p=\varprojlim_k\mathbb{Z}/p^k\mathbb{Z}.$

The full profinite completion combines all primes at once:

$\displaystyle \widehat{\mathbb{Z}}\cong\prod_p\mathbb{Z}_p.$

So Furstenberg’s topology is not an isolated curiosity. It is an elementary entrance into the same world as $p$ -adic numbers, profinite groups, local-global principles, and arithmetic geometry.

One useful application of this viewpoint is the concept of density. A subset $A\subseteq\mathbb{Z}$ is dense in the profinite topology if it meets every nonempty congruence class. That means: for every modulus $n$ and every residue class $b\pmod n$ that is allowed, there is some element of $A$ congruent to $b$ modulo $n$ .

Dirichlet’s theorem on primes in arithmetic progressions can be viewed in this language. The theorem says that if $\gcd(a,n)=1,$ then the arithmetic progression $a+n\mathbb{Z}$ contains infinitely many primes. Topologically, this says that the primes are dense in the unit part of the profinite integers. They do not meet every congruence class, because a prime cannot lie in a residue class that is forced to have a common factor with the modulus. But among the reduced residue classes, the primes appear everywhere. This topological phrasing does not prove Dirichlet’s theorem. The proof of Dirichlet’s theorem requires deeper analytic number theory. But the topology tells us what the theorem really means: primes are not merely infinite; they are spread throughout the congruence topology of the integers. They are dense in the space of possible residue classes compatible with being prime. This distinction is important. Furstenberg’s proof uses topology to prove a structural infinitude result. But deeper distribution results usually require more than topology. Topology gives the correct language: density, closure, open sets, finite quotients.

The same perspective becomes central in Galois theory. For a finite Galois extension $L/K$ , the Galois group ${\text{Gal}}(L/K)$ is a finite group. But if we take a large infinite extension, such as the separable closure $K^{\mathrm{sep}}$ of a field $K$ , then the absolute Galois group

$\displaystyle {\text{Gal}}(K^{\mathrm{sep}}/K)$

is not just an abstract group. It is naturally a profinite group:

$\displaystyle {\text{Gal}}(K^{\mathrm{sep}}/K)=\varprojlim_{L/K}{\text{Gal}}(L/K),$

where $L$ runs over the finite Galois extensions of $K$ inside $K^{\mathrm{sep}}$ . This is the same idea again; understand an infinite arithmetic object through all of its finite quotients. The topology on the absolute Galois group is called the Krull topology. A neighborhood of the identity consists of automorphisms that act trivially on some finite extension. Thus two Galois automorphisms are close if they agree on a large finite piece of the algebraic closure. This is directly parallel to the topology on $\mathbb{Z}$ . Two integers are close if they agree modulo a finite quotient $\mathbb{Z}/n\mathbb{Z}$ . Two Galois automorphisms are close if they agree after passing to a finite Galois quotient. In both cases, closeness means indistinguishability by finite tests. This profinite Galois viewpoint is one of the pillars of modern number theory. Primes are connected to Galois groups through Frobenius elements. Roughly speaking, given a finite Galois extension of number fields, most prime ideals determine conjugacy classes in the Galois group. The Chebotarev density theorem says that these Frobenius conjugacy classes are distributed evenly among the conjugacy classes of the Galois group. Dirichlet’s theorem on primes in arithmetic progressions is a special abelian case of this general principle. Thus the same kind of topological language that begins with Furstenberg’s proof eventually leads to a much larger picture: finite congruence data leads to profinite completions; profinite completions lead to profinite Galois groups; profinite Galois groups organize the distribution of primes.

The idea also appears in group theory. Given an abstract group $G$ , one can form its profinite completion

$\displaystyle \widehat G=\varprojlim_{N}G/N,$

where $N$ runs over finite-index normal subgroups of $G$ . This construction records all possible finite quotients of $G$ at once. A group is called residually finite if every nontrivial element can be detected in some finite quotient. In topological language, this means the natural map

$\displaystyle G\to\widehat G$

is injective. Again the same question appears: what can finite quotients see? In Furstenberg’s proof, finite congruence quotients cannot see enough primes if the list of primes is finite. In group theory, finite quotients may or may not distinguish elements, subgroups, or even whole groups. Profinite topology gives a precise language for these questions.

There is also a geometric version in algebraic geometry. In ordinary topology, the fundamental group classifies covering spaces. In algebraic geometry, the natural covering spaces are finite étale covers. Because the relevant covers are finite, the corresponding fundamental group is not an ordinary discrete group but a profinite group. This is the étale fundamental group. Once again, finite algebraic data is organized into an inverse limit, and the resulting object is topological.

So Furstenberg’s proof is a small door into a much larger world. Its immediate purpose is to prove that there are infinitely many primes. But its deeper lesson is methodological. It shows that if we choose the right topology, arithmetic statements can become geometric statements. Divisibility becomes closedness. Congruence becomes openness. Finite congruence information becomes local information. Infinite arithmetic behavior becomes behavior at the level of closure, density, or compactness. The proof is therefore not merely saying, Here is a simple topological proof of Euclid’s theorem. It is saying something more general: Arithmetic can be studied by asking what is visible through all finite quotients. For the integers, the finite quotients are $\mathbb{Z}/n\mathbb{Z}.$ For a group, the finite quotients are $G/N.$ For an absolute Galois group, the finite quotients are the Galois groups of finite extensions. For algebraic varieties, the finite quotients come from finite étale covers. In each case, topology provides the structure that lets all finite information be held together at once.

This is why the notion of closeness is so important. The topology decides what kind of information matters. In the ordinary topology on the integers, closeness means small numerical difference. In the Furstenberg topology, closeness means congruence similarity. In the $p$ -adic topology, closeness means divisibility of the difference by a high power of $p$ . In the profinite topology on a group, closeness means agreement in finite quotients. In Galois theory, closeness means agreement on finite extensions.

The final message is this. Furstenberg’s proof is not important because it makes the infinitude of primes easier. Euclid’s proof is already easier. It is important because it teaches a way of thinking. It teaches us to treat congruence conditions as neighborhoods, finite quotients as observations, and arithmetic limits as topological limits. This way of thinking leads naturally to profinite integers, $p$ -adic analysis, Galois groups, Chebotarev-type distribution theorems, residual finiteness in group theory, and étale fundamental groups in algebraic geometry. In this sense, the proof is a miniature model of a major modern theme: to understand an infinite arithmetic object, study all of its finite shadows, and then give those shadows a topology.

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