Euclid's theorem
theorem that the number of prime numbers is infinite
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements. There are at least 200 proofs of the theorem.
Euclid's proof
Euclid offered a proof in his work Elements (Book IX, Proposition 20), which is paraphrased here.
Consider any finite list of prime numbers p1, p2, ..., pn. It will be shown that there exists at least one additional prime number not included in this list. Let P be the product of all the prime numbers in the list: P = p1p2⋅⋅⋅pn. Let q = P + 1. Since q is either prime or not:
If q is prime, then there is at least one more prime that is not in the list, namely, q itself.
If q is not prime, then some prime factor p divides q.
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