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I've passed by this article: http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/

and this paper: http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf

They say that there is a formula such that when you give it (n) then it returns the n-th prime number. Where other articles states that no formula discovered so far that does such thing.

If the formula exists indeed, then why from time to time they discover a new largest prime number known ever. It would be very simple using the formula to find a larger one.

I just want to ensure whether such formula exists or not.

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Actually, most of these "formulas" are based on Sieve theory. –  Shane Chern Mar 17 '13 at 8:40
    
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up vote 2 down vote accepted

As you already mention yourself: it doesn't make sense to keep on looking for prime numbers with computer algorithms if there is a prime number equation.

Looking at the formulas on the site you provided, it seems to me that the formulas are really just an algorithm which allows you to determine whether some number is a prime number based on the previously found prime numbers. That I could already do in highschool by just checking whether a number can be divided by the previous prime numbers with integer solution (larger then 1). Probably, I cannot judge that quickly, the algorithm is more efficient then what I mention, but it still not a 'plug in the numbers and have your answer with a pocket calculator'.

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It depends on what you mean by "formula". Certainly no formula is known such that using it to find a new very large prime would be "very simple".

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If formulae for computing primes existed, there wouldn't be a a thing called 'Largest known primes' . And moreover, there are few primes like Mersenne primes and Fermat Prime. But eventually, their converse isn't true ALWAYS.

For eg: Mersenne Prime.

$q$ is $prime$ which is equal to $2^p-1$, this shows $p$ is a prime. But the converse, take a prime $a$, it is always not true that $2^a-1$ is a prime.

Similarily for the Fermat primes. So, there are no thing called 'Formula' to find primes.

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