How did Euclid prove that infinitely many prime numbers exist?

In his work Elements, Euclid proved that there are infinitely many prime numbers using a logical argument found in Proposition 20 of Book IX. This proof demonstrates that any finite collection of primes can always be expanded, meaning no finite set can contain all of them.1

Euclid’s argument begins by considering a finite set of primes, such as $a,b,c,\dots ,n$. He then constructs a new number, $N$, by multiplying these primes together and adding one:
$$N=(a\times b\times c\times \dots \times n)+1$$

Euclid then examined two possibilities for $N$:

• If $N$ is prime: Since $N$ is larger than any of the primes in the original set, it must be a new prime not included in that collection.1
• If $N$ is composite: It must have a prime factor. Euclid demonstrated that this prime factor cannot be any of the primes in the original set, as dividing $N$ by any of those primes would always leave a remainder of 1.1

In either case, a new prime is identified, proving that the collection of prime numbers is infinite.1