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If you plot the ratios of consecutive primes against the number of preceding primes there is a whole bunch of distinct lines.

Do they have a name? Do the individual lines have a formula?

Here's a plot of $p_n/p_{n-1}$ against $n$ where $p_n$ is the $n$th prime. p_n/p_{n-1}

Lucas
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  • Maybe you can have a look here and here for a clue... – AugSB Nov 14 '17 at 23:50
  • Matthew's deleted answer is almost good : you are plotting $(n+1,1+\frac{g(n)}{p_n})\approx (n+1,1+\frac{g(n)}{n \log n})$ where $g(n)= p_{n+1}-p_n$ is an integer and $p_n \sim n \log n$ is the prime number theorem. The distinct curves are the different values of $g(n)$. – reuns Nov 15 '17 at 02:29
  • @reuns Yeah, I plotted functions of that form, they're close, but what was odd is that they were out by a really consistent amount. I added in a term of $\log\log n - 1$ and that seems to have fixed it. – Lucas Nov 15 '17 at 08:44
  • $\log \log n-1$ where ? – reuns Nov 15 '17 at 21:38
  • @reuns I used the approximation $p_n = n (\log n + \log\log n - 1)$ – Lucas Nov 16 '17 at 07:51

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