Euclid's theorem

Euclid's theorem states that there are infinitely many prime numbers. Its classic proof assumes a finite list of primes and constructs an integer not divisible by any prime on that list.

Euclid's proof of the infinitude of primes: A proof that assumes a complete list of primes and considers one plus their product. This is the classic argument establishing the theorem.
Euclid: An ancient Greek mathematician whose surviving work shaped the later study of geometry and number theory. The theorem is traditionally attributed to him.
Prime number: A natural number greater than one whose only positive divisors are one and itself. The theorem concerns the size of this set of numbers.
Euler's proof of the infinitude of primes: A proof that uses the divergence of the sum of reciprocals of primes to show there are infinitely many primes. It replaces Euclid's finite-list contradiction with an argument about infinite series.
Euclid–Mullin sequence: A sequence formed by repeatedly taking the least prime factor of one plus the product of earlier terms. It iterates the theorem's product-plus-one construction to generate primes.
Elements: Euclid's mathematical treatise, organized into thirteen books covering geometry, number theory, and related topics. Book IX contains the classic proof that primes are infinite.
Divisibility: An integer divides another when the latter equals the former multiplied by an integer. The proof tests whether listed primes divide the constructed number.
Furstenberg's proof of the infinitude of primes: A proof of prime infinitude using a topology on the integers generated by arithmetic progressions. It reaches the same conclusion through a surprising connection to topology.
Fundamental theorem of arithmetic: Every integer greater than one has a unique factorization into primes, apart from the order of factors. The constructed integer must have a prime divisor, even if it is not itself prime.
Greek mathematics: Mathematical traditions developed in the Greek-speaking world from antiquity through the Hellenistic period. Euclid's proof belongs to this historical tradition.

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