 Euclid's theorem

There are infinitely many prime numbers. Indeed, for any finite list of prime numbers $p_1,\dots ,p_k$, the integer $N=p_1\cdots p_k+1>1$ has a prime factor, by the fundamental theorem of arithmetic. None of the listed prime numbers divides $N$, since each divides $N-1$ and a common divisor would divide $1$. Thus no finite list contains all prime numbers. The constructed $N$ need not itself be a prime number.

For example, the first six prime numbers giveBoth prime factors lie outside the original list, although $30031$ is a composite number. The construction defeats a proposed complete list; it is not a formula guaranteed to produce a prime number. Book IX, Proposition 20 of Euclid's Elements explicitly considers both possibilities, prime and composite. ^[ref].