**A proof that there are infinitely many prime numbers**

Suppose there are a finite number of primes, say N.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

None of the primes p_1...p_N divides S.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

None of the primes p_1...p_N divides S.

So either S is itself prime,

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

None of the primes p_1...p_N divides S.

So either S is itself prime,

or S is divisible by some prime not on our list.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

None of the primes p_1...p_N divides S.

So either S is itself prime,

or S is divisible by some prime not on our list.

In either case there is a prime number not yet on our list.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

None of the primes p_1...p_N divides S.

So either S is itself prime,

or S is divisible by some prime not on our list.

In either case there is a prime number not yet on our list.

So our initial assumption must be false.

Suppose there are a finite number of primes, say N.

Then we can list these as p_1, p_2,..., p_N.

Form the number S = (p_1 x p_2 x p_3...x p_N) + 1.

None of the primes p_1...p_N divides S.

So either S is itself prime,

or S is divisible by some prime not on our list.

In either case there is a prime number not yet on our list.

So our initial assumption is false.

We conclude there are infinitely many primes.

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