- #1
22-16
[SOLVED] Are prime numbers infinite?
Are prime numbers infinite[?] [?] [?]
Are prime numbers infinite[?] [?] [?]
Do Prime Numbers Continue Indefinitely?
- Context: High School
- Thread starter 22-16
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Discussion Overview
The discussion centers around the question of whether prime numbers are infinite, exploring both theoretical aspects and implications of prime numbers in various contexts, including mathematical proofs and applications in encryption.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants assert that there are an infinite number of primes, referencing a proof attributed to Euclid that involves constructing a number that cannot be divisible by any finite list of primes.
- Others engage in a discussion about the grammatical correctness of phrasing related to the infinitude of primes, indicating a focus on clarity in communication.
- One participant mentions that the gaps between prime numbers increase as numbers grow larger, suggesting a perceived randomness in the distribution of primes.
- Another participant introduces the application of prime numbers in encryption, noting the difficulty of factoring large numbers into their prime components.
- A later reply raises the question of whether there are an infinite number of prime pairs, indicating further exploration of the topic.
Areas of Agreement / Disagreement
Participants generally agree that there are an infinite number of primes, but the discussion includes varying perspectives on the implications and characteristics of prime numbers, particularly regarding their distribution and applications. The question of prime pairs remains unresolved.
Contextual Notes
The discussion includes assumptions about the definitions of prime numbers and the implications of their infinitude, but these assumptions are not explicitly stated or resolved. The exploration of prime pairs introduces additional complexity that is not fully addressed.
Who May Find This Useful
Readers interested in number theory, mathematical proofs, cryptography, and the properties of prime numbers may find this discussion relevant.
- #2
KLscilevothma
- 315
- 0
Yes, there are an infinite number of primes
KL Kam
KL Kam
- #3
KLscilevothma
- 315
- 0
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"
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
Science Advisor
Homework Helper
- 42,895
- 983
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.)
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
KLscilevothma
- 315
- 0
HallsofIvy, thanks for giving me an English lesson under the topic "Are prime numbers infinite?"
- #6
Ed Quanta
- 296
- 0
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.
- #7
FrankM
- 26
- 0
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.
- #8
thed
But the million dollar question is, are there an infinite number of prime pairs?
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