What is the recursive formula for prime numbers?

Question

Like recursive formula for this number series is $L=\{1,5,9,13,17,...\}$
let $T(1)=1$; then formula for all series is $T(N)=T(N-1)+4$; then what will be the generic formula for prime numbers?

Answer 1

$p(N) = \min(k>p(N-1), \forall n<N, k \mod p(n)=0)$

is the form given by the definition of prime numbers, unfortunately it doesn't provide a fast way to compute the $n$-th prime number.

Answer 2

Let $p_1 = 2$ and let $p_n$ be the smallest number greater than 1 which is not divisible by any of $p_1, \ldots, p_{n-1}$.

Answer 3

You may want to read this paper. It gives something which is recursive but not useful. I found it very nice though!!https://arxiv.org/pdf/1607.05267.pdf

Answer 4

I want to show what I found today: https://arxiv.org/abs/0711.3940

You can not believe in everything you found in arxiv.org. Still this is an interesting result and what you are looking for.