## question on straigh forward calculation of prime numbers

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• Hi, I have a question where I need a very very reliable answer: Does there exist any function in the world to which I can input the first 6 prime numbers and
Message 1 of 2 , Mar 7, 2013
Hi,

I have a question where I need a very very reliable answer:

Does there exist any function in the world to which I can input the first 6 prime numbers and by finding its zeros I could just get the next 12768 prime numbers - and so forth?

I am indeed talkig about a straight forward calculation and not a sieving etc.

I would be deeply thankful for your help.

-Sam
• ... Did you google ? There are explicit formulae for prime(n) but they are inefficient to compute. For example, on
Message 2 of 2 , Mar 7, 2013
On Thu, Mar 7, 2013 at 10:55 AM, viva8698 <vaseghisam@...> wrote:
> Hi,
>
> I have a question where I need a very very reliable answer:
>
> Does there exist any function in the world to which I can input the first 6 prime numbers and by finding its zeros I could just get the next 12768 prime numbers - and so forth?

There are explicit formulae for prime(n)
but they are inefficient to compute.
For example, on http://en.wikipedia.org/wiki/Formula_for_primes
you find
p_n = 1 + \sum_{k=1}^{2(\lfloor n \ln(n)\rfloor+1)} \left(1 -
\left\lfloor{\pi(k) \over n} \right\rfloor\right)
with
\pi(k) := k - 1 + \sum_{j=1}^k \left\lfloor {2 \over j} \left(1 +
\sum_{s=1}^{\left\lfloor\sqrt{j}\right\rfloor} \left(\left\lfloor{ j-1
\over s}\right\rfloor - \left\lfloor{j \over s}\right\rfloor\right)
\right)\right\rfloor

Now these aren't directly given as "roots of polynomials"
even though that could probably be done ;

OTOH I don't understand really well your "and so forth" - can you give
an example for any other sequence of numbers, to see how that should
work ?

Maximilian

>
> I am indeed talkig about a straight forward calculation and not a sieving etc.
>
> I would be deeply thankful for your help.
>