- #1

#### 22-16

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

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

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In summary, the conversation discusses the topic of whether prime numbers are infinite. The first post presents a proof by contradiction that there are infinitely many primes. The following posts discuss the ambiguity of the original question and provide additional information about prime numbers and their applications. Overall, the conclusion is that there are indeed an infinite number of primes, and their distribution appears to be random. The question of whether there are an infinite number of prime pairs remains unanswered.

- #1

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

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- #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

Science Advisor

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

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- #8

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

Prime numbers are positive integers that are only divisible by 1 and themselves. Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

This is a well-studied and proven theorem in mathematics known as Euclid's Theorem. It states that there are infinitely many prime numbers, and this has been confirmed through various mathematical proofs and experiments.

While there is no known general pattern for prime numbers, there are some interesting patterns and properties that have been discovered. For example, primes are always odd (except for 2), and they become less frequent as we move to larger numbers.

If prime numbers are not infinite, it would have a significant impact on the field of mathematics. Many mathematical proofs and algorithms rely on the assumption that there are infinitely many prime numbers. Without this assumption, the validity of these proofs would be called into question.

As of now, the largest known prime number is 2^82,589,933 - 1, which has over 24 million digits. However, there is no definitive answer to whether there is an absolute largest prime number or not. As prime numbers are infinite, there is always a possibility of discovering a larger prime number in the future.

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