- #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
- Start date
Click For Summary
SUMMARY
Prime numbers are infinite, as established by KL Kam's proof, which is rooted in Euclid's original argument. The proof assumes a finite number of primes and leads to a contradiction by constructing a new number, Q, that cannot be divided by any of the assumed primes. This discussion also touches on the increasing gaps between prime numbers and their randomness, as well as their application in encryption, where large prime factors are challenging to compute.
PREREQUISITES- Understanding of basic number theory concepts
- Familiarity with Euclid's proof of the infinitude of primes
- Knowledge of prime factorization and its applications in cryptography
- Basic programming skills for implementing algorithms related to prime factorization
- Study Euclid's proof of the infinitude of primes in detail
- Explore algorithms for prime factorization, such as the Pollard rho algorithm
- Research the distribution of prime numbers and the Prime Number Theorem
- Learn about cryptographic systems that utilize prime numbers, such as RSA encryption
This discussion is beneficial for mathematicians, computer scientists, cryptographers, and anyone interested in the properties and applications of prime numbers.
- #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?
Similar threads
- · Replies 5 ·
- · Replies 17 ·
- · Replies 8 ·
- · Replies 5 ·
- · Replies 1 ·
High School
Aren't There Any Formulas for Prime Numbers?
- · Replies 7 ·
- · Replies 58 ·
- · Replies 1 ·
- · Replies 1 ·
- · Replies 8 ·