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22-16

**[SOLVED] Are prime numbers infinite?**

Are prime numbers infinite[?] [?] [?]

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22-16

Are prime numbers infinite[?] [?] [?]

- #2

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Yes, there are an infinite number of primes

KL Kam

KL Kam

- #3

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assume there exist only finite number of primes, say p1, ... ,pn

Consider Q=p1 ... pn + 1

is Q a prime number?

If yes, this means that there exist a prime other than p1 ... pn (absurd!)

is Q composite?

now Q is not divisible by pi , then Q must contains divisors other than p1 ... pn

The result follows.

PS Grammar mistake in my last post, it should be "there are infinite number of primes"

- #4

HallsofIvy

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You are, of course, completely correct in calling attention to the fact that the original question "are prime numbers infinite" is ambiguous and rephrasing it.

(Oh, by the way, your proof that there are an infinite number of primes is certainly completely correct and goes back to Euclid himself.)

- #5

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No, you first post "there are AN infinite number of primes" was grammatically correct. "There are infinitely many primes" would also be correct. "There are infinite number of primes" is not grammatically correct.

HallsofIvy, thx for giving me an English lesson under the topic "Are prime numbers infinite?"

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- #8

thed

But the million dollar question is, are there an infinite number of prime pairs?

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