Prime number

Two, three, five and seven are prime; four, six, eight and nine are composite. One is neither. Book VII separates the unit from numbers, a historical vocabulary requiring care when compared with modern integer categories.

Two numbers are relatively prime when their only positive common divisor is one. Eight and nine are relatively prime although both are composite. A prime paired with itself is not a relatively prime pair.

References: [1]

VII.30 establishes that a prime dividing a product ab divides at least one factor. Seven divides fourteen times fifteen because it divides fourteen; it need not divide both factors.

Replacing the prime with an arbitrary composite fails. Six divides two times three but divides neither factor. Divisibility of a product and divisibility of an individual factor are distinct conditions.

References: [2]

IX.20 shows that no finite list exhausts the primes. Take a positive common multiple M of the listed primes and consider M plus one. None of the listed primes divides it, since such a divisor would also divide their difference, one.

M plus one may itself be prime or composite. In the latter case a prime divisor still lies outside the list. The argument requires a new prime divisor, not the false claim that every product of primes plus one is prime.

References: [3]