Euclid's theorem
= Euclid's theorem
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There are infinitely many <prime numbers>. Indeed, for any finite list of <prime numbers> $p_1,\ldots,p_k$, 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>.