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Theorem 2.3 (Infinitude of primes) There are infinitely many prime numbers. PROOF: Suppose you have a finite list of prime numbers. Multiply all the prime numbers in your list together, and call the result N. So N is a positive integer. As N + 1 is again a positive integer, bigger than 1, N + 1 has a prime factor p. Remember, you began with a list of prime numbers! If p were in your list, then p | N (since N was the product of primes in your list) and p | N + 1 (we chose p to be a factor of N + 1). By the two-out-of-three principle, p | ((N + 1) - N), so p | 1. But no prime number is a factor of 1! Hence the hypothesis ("if p were in the list") cannot be true. The prime number p is a "new prime", a prime not in the list. Since no finite list of prime numbers contains all the prime numbers, there are infinitely many prime numbers.

          Theorem 2.3 (Infinitude of primes) There are infinitely many prime numbers.

PROOF: Suppose you have a finite list of prime numbers. Multiply all the prime numbers in your list together, and call the result N. So N is a positive integer. As N + 1 is again a positive integer, bigger than 1, N + 1 has a prime factor p.
Remember, you began with a list of prime numbers! If p were in your list, then p | N (since N was the product of primes in your list) and p | N + 1 (we chose p to be a factor of N + 1). By the two-out-of-three principle,
p | ((N + 1) - N), so p | 1.
But no prime number is a factor of 1! Hence the hypothesis ("if p were in the list") cannot be true. The prime number p is a "new prime", a prime not in the list.
Since no finite list of prime numbers contains all the prime numbers, there are infinitely many prime numbers.

Theorem 2.3 (Infinitude of primes) There are infinitely many prime numbers.

PROOF: Suppose you have a finite list of prime numbers. Multiply all the prime numbers in your list together, and call the result N. So N is a positive integer. As N + 1 is again a positive integer, bigger than 1, N + 1 has a prime factor p.
Remember, you began with a list of prime numbers! If p were in your list, then p | N (since N was the product of primes in your list) and p | N + 1 (we chose p to be a factor of N + 1). By the two-out-of-three principle,
p | ((N + 1) - N), so p | 1.
But no prime number is a factor of 1! Hence the hypothesis ("if p were in the list") cannot be true. The prime number p is a "new prime", a prime not in the list.
Since no finite list of prime numbers contains all the prime numbers, there are infinitely many prime numbers.

Added by Lisa G.

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