Do Prime Numbers Continue Indefinitely?

AI Thread Summary
The discussion confirms that there are indeed an infinite number of prime numbers, supported by a proof attributed to Euclid. The proof involves assuming a finite number of primes and demonstrating that this leads to a contradiction when considering a number formed by multiplying all known primes and adding one. The conversation also touches on the grammatical nuances of discussing primes, with participants correcting each other on phrasing. Additionally, it highlights the increasing gaps between prime numbers as they grow larger and notes the randomness in their distribution. The practical application of prime numbers in encryption is mentioned, emphasizing the difficulty of factoring large numbers into their prime components, which poses challenges even for advanced computational systems. The thread concludes with a curiosity about the existence of infinite prime pairs.
22-16
[SOLVED] Are prime numbers infinite?

Are prime numbers infinite[?] [?] [?]
 
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Yes, there are an infinite number of primes

KL Kam
 
Proof:
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" :smile:
 
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.

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.)
 
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?" :wink:
 
I will go on and give the reassuring answer of yes. And the amount of numbers in between prime numbers increases as the numbers increase, and the pattern of prime numbers appears to be completely random.
 
Even more useless information about primes. Some encryption codes use the multiple of two primes. Since really big numbers are very time consuming to factor, even super computers (if there is such a thing anymore) needs days weeks months even years to factor the number into the original two primes. Read that in scientific american I think. I had a lot of fun for a few days trying to write code that would factor these numbers really fast (or even not so fast) but got absolutly nowhere.
 
But the million dollar question is, are there an infinite number of prime pairs?
 

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