Proposition IX.20 — Prime numbers are more than any assigned multitude of prime …

Proof

Given primes

p_{1}, \dots, p_{n}

, form

N = p_{1} p_{2} \dots p_{n} + 1

. By VII.31

N

has a prime divisor

q

. If

q

were one of the

p_{i}

, then

q

would divide

N - p_{1} \dots p_{n} = 1

(Common Notion 3), which is impossible. Hence

q

is a new prime not in the original list; the list of primes admits no upper bound.

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Full neighborhood

Depends on (3)

VII.31Proposition VII.31Any composite number is measured by some prime number.
3Common notion 3If equals be subtracted from equals, the remainders are equal.
VII.11Definition VII.11A prime number is that which is measured by a unit alone.

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