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I have discovered a recursive formula for the density of the primes, and would like to ask whether this is novel or is already known, or perhaps equivalent to something else.

Let $ P_n$ denote the nth prime

My proof shows that the density of primes in the number range $ P_n^2 < x < P_{n+1}^2$ is given by

$D_n = \frac{P_n - 1}{P_n} . D_{n-1} $

where seed values are

$D_0 = 1$

$P_1 = 2$

Is this useful to anyone?

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Something is very wrong here.If any formula of this shape was correct, it would give a fast way to calculate $\pi(x)$. Check it for not too small values of $n$... – Feldmann Denis Jan 7 at 4:35
It is a nice approximation, but there are some small discrepancies with the actual ratios. You might compute the discrepancies and report back. It may be the difference is small enough that your approximate formula will still be useful. Gerhard "Also Interested In Prime Differences" Paseman, 2013.01.06 – Gerhard Paseman Jan 7 at 5:04
I get $D_3 = 4/15$ by your formula, but the actual density should be $6/23$. Through small values of $n$, I find that the actual density is about $0.01$ below your formula, but this is the sort of thing that small $n$ are misleading about. – Kevin O'Bryant Jan 7 at 5:24
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 If you change "proof" to "argument", and change the question to "are there infinitely many $n$ where this underestimates the true density", and you'll avoid a closed question. – Kevin O'Bryant Jan 7 at 5:34

closed as off topic by Felipe Voloch, Andy Putman, Andres Caicedo, Will Jagy, Igor Pak Jan 7 at 7:32

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