Does anyone know any information on the bands appearing in ratios of consecutive prime numbers? For example, if any analytic explicit or recursive formulas exist per band. In other terms, how do I compute points of a specific band? Anything relevant at all is appreciated. Code in the Wolfram Language for the first 5000 ratios:


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    $\begingroup$ These almost certainly correspond to the 'prime gaps': The bottommost corresponds to $p_{n+1}=p_n+2$, or in other words $\frac{p_{n+1}}{p_n}=1+\frac2{p_n}$; the next one to a ratio of $1+\frac4{p_n}$; etc. $\endgroup$ Jan 10 '21 at 21:57
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    $\begingroup$ To expand on the previous comment: what's going on is more clear if you plot differences instead of quotients: ListPlot[Prime[# + 1] - Prime[#] & /@ Range[5000]] You'll see a band for every even number - which are, of course, the only numbers that could appear as differences. $\endgroup$ Jan 10 '21 at 22:07

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