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Lets say that we have the set $P = { 2, 3, 5, 7, ... }$, where $P$ is the set of all prime numbers ...

The series $P_1 + P_2 + P_3 + ...$ does it converge or diverge ..?

$\sum_{P_i \in P} P_i \; = \; ?$

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Think. If that fails, plot.

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figure: Spiral-Plot of prime numbers under 2026

I think I can't get it, I'll do the plots later ...

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Primes are an infinite set of positive integers, and zero is not prime.

The series diverges.

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The series diverges.

rapidly

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The series diverges.

rapidly

Is that enough to say: $\sum_{P_i \in P} P_i = \infty$ ..?

Because, besides Euclid's work, all formula that can be used to generate prime numbers create series that diverges ...

Edited by khaled
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figure: Spiral-Plot of prime numbers under 2026

I think I can't get it, I'll do the plots later ...

What you posted is Sacks spiral, the same as Ulams spiral except using an Archimedean spiral instead of a square one, and has nothing whatsoever to do with sums of primes. In any case, what I meant by plot was $f(x) = \sum_{i=1}^{x} P_{i}$.

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Is that enough to say: $\sum_{P_i \in P} P_i = \infty$ ..?

Because, besides Euclid's work, all formula that can be used to generate prime numbers create series that diverges ...

That is to say that $\lim_{n \to \infty} P_n = \infty$ so that to think that the sum might actually converge is absurd.

You are considering an "infinite sum" of positive integers, with the individual terms growing rapidly -- which is obvious from the fact that there are infinitely many primes and more so from the prime number theorem.

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That is to say that $\lim_{n \to \infty} P_n = \infty$ so that to think that the sum might actually converge is absurd.

I agree and was wondering if the original poster was simply confused about the meaning of converging and diverging.

i got it

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Furthermore, is there an equation than can express the nth value of any prime number? For example, where n=1, F(n)=2 etc.

Nope

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Furthermore, is there an equation than can express the nth value of any prime number? For example, where n=1, F(n)=2 etc.

No

In fact no simple expression that produces only prime numbers is known, let alone one that would list all of them in increasing order.

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No

In fact no simple expression that produces only prime numbers is known, let alone one that would list all of them in increasing order.

That is probably the Holy Grail of Number Theory, and one that we all suspect is an impossibility.

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