Infinitude of Primes - A Topological Proof without Topology

Infinitude of Primes
A Topological Proof without Topology

Using topology to prove the infinitude of primes was a startling example of interaction between such distinct mathematical fields is number theory and topology. The example was served in 1955 by the Israeli mathematician Harry Fürstenberg. More than 50 years later, Idris D. Mercer has recast the proof into more algebraic terms, eschewing the language of topology.

The proof in this form loses some of the charm that made Fürstenberg's proof a good candidate for The Book envisaged by P. Erdös, but it also shows how naturally topological ideas permeate other branches of mathematics.

Following Fürstenberg we consider the sets of two-sided arithmetic progressions (AP, for short).
N_{a, b} = { a + nb: n ∈Z }

and abbreviate

NM(m) = N_{1, m} ∪ N_{2, m} ∪ ... ∪ N_{m-1, m},

so that NM(m) denotes the set of numbers not divisible by m.

Lemma 1

A finite intersection of APs is either empty or infinite.

Proof

Given a finite set of APs, N_{a_k, b_k}, k = 1, ..., s, the intersection ∩N_{a_k, b_k} of all N_{a_k, b_k}'s is either empty or contains an integer, say, x. In the latter case, x is a member of all APs N_{a_k, b_k}, k = 1, ..., s. But then the same is true for x + y, where y is any common multiple of b_k, k = 1, ..., s, proving the lemma.

(More accurately we may claim that x ∈ N_{x, b₁b₂ ... b_s}. In topological terms this shows that the intersection of a finite number of open neighborhoods of x is an open neighborhood.)

Lemma 2

If S is any collection of sets, then a finite intersection of finite unions of members of S is also a finite union of finite intersections of members of S.

Proof

This is a consequence of de Morgan's Formulas.

Theorem

There are infinitely many primes.

Proof

If p₁, ..., p_t are all the primes there are then

NM = NM(p₁) ∩ ... ∩ NM(p_t) = {-1, 1};

for any integer, except for -1 and 1, is the product of a number of primes. NM is a finite intersection of finite unions of APs. Hence, by Lemma 2, it is also a finite union of finite intersections of APs. By Lemma 1, each of this intersections is either empty or infinite and the same holds for their finite unions. This contradicts the fact that NM contains only 2 elements -1 and 1.

Reference

M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 2000
I. D. Mercer, On Furstenberg's Proof of the Infinitude of Primes, Amer. Math. Monthly 116, n 4 (April 2009), 355

Infinitude of Primes

71546956