# Prime numbers searching

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?

Hurkyl
Staff Emeritus
Gold Member
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"

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hypnagogue
Staff Emeritus
Gold Member
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?

Hurkyl
Staff Emeritus
Gold Member
See for yourself:

http://mathworld.wolfram.com/PrimeFormulas.html

Summing over 2^n terms becomes inefficient really quick. 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 recieved a rigorus lashing on how math isn't relegion. Ha!