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## Main Question or Discussion Point

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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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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Hurkyl

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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"

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

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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?Originally posted by Hurkyl

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

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Hurkyl

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

Summing over 2^n terms becomes inefficient really quick.

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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!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?

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