Exploring the Infinity of Prime Numbers: The Cardinality Conjecture

In summary, there is an explicit formula for the n-th prime number, the prime numbers are countably infinite, and the smallest infinite cardinal is aleph-0, as there exists a bijection with the Natural Numbers. There is also a known limit to the largest prime number due to limitations in computational resources.
  • #1
kerimek
1,385
0
Does exist any proof that prime numbers cannot be generated sequentially without jump across any one? And which is cardinality of prime numbers set? Is the set "the smallest" infinite set?
 
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  • #2
I'm not entirely sure what you mean...


There does actually exist an explicit (but complicated) formula for the n-th prime number.

The prime numbers are countably infinite, and that is the smallest infinite cardinal. (However, the integers, rationals, and even the algebraic numbers are each countably infinite as well)


edit: fixed the omission of the word "infinite" from "smallest infinite cardinal"
 
Last edited:
  • #3
Originally posted by Hurkyl
There does actually exist an explicit (but complicated) formula for the n-th prime number.

If this is so, why does there exist a number that is called "the largest known prime number"? Limitations of computational resources I'm guessing?
 
  • #5
Originally posted by kerimek
Does exist any proof that prime numbers cannot be generated sequentially without jump across any one? And which is cardinality of prime numbers set? Is the set "the smallest" infinite set?

The cardinality of the prime numbers is aleph-0, there exists a bijection with the Natural Numbers. I once sugested this exact conjecture with an old professor of mine and received a rigorus lashing on how math isn't relegion. Ha!
 

1. What are prime numbers?

Prime numbers are positive integers that are divisible only by 1 and themselves. They have exactly two factors, 1 and the number itself.

2. Why is it important to search for prime numbers?

Prime numbers have many important applications in fields like cryptography, computer science, and mathematics. They are also used in algorithms for solving various problems.

3. How do scientists search for prime numbers?

There are various methods for searching for prime numbers, such as using sieves, algorithms, and mathematical formulas. Some commonly used algorithms include the Sieve of Eratosthenes and the AKS primality test.

4. Can there be an end to the search for prime numbers?

No, the search for prime numbers is an ongoing process as there is no limit to the number of prime numbers. As numbers get larger, it becomes more difficult to find new primes, but there will always be more to discover.

5. Are there any practical applications of prime number searching?

Yes, prime numbers have many practical applications, such as in encryption algorithms used to secure online transactions and protect sensitive information. They are also used in creating unique identification numbers and in generating random numbers for various applications.

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