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April 22nd, 2019, 01:36 PM  #1 
Member Joined: Feb 2013 From: London, England, UK Posts: 38 Thanks: 1  Prime Numbers and the relationship between n and P_n
We often hear it said that either there is no relation between the natural, or counting numbers, $n$, and their counterparts the primes, $P_{n}$, or that if there is, it is so recondite as to be completely obscure. This is something of a sacred cow, but I don't believe it to be true. I'm not claiming the following is an exact relationship by any means, but it is a start, and a step in the right direction; it holds pretty well for the first 100,000 primes with remarkable accuracy: 2*$n$*$P_{n}$ $\approx$ $P_{n^2}$ Examples: $n$$P_{n}$2*$n$*$P_{n}$$P_{n^2}$ 1242 23127 353023 475653 51111097 613156151 717238227 819304311 923414419 1029580541 1131682661 1237888827 134110661009 144312041193 154714101427 165316961619 175920061879 ... ... ... 773895990658579 783976193260289 794016335862081 804096544063809 ... ... ... 
April 22nd, 2019, 07:53 PM  #2 
Senior Member Joined: Mar 2019 From: iran Posts: 318 Thanks: 14 
2nP(n) = 2n*nln(n) = 2n^2ln(n) = n^2 ln(n^2) = P(n^2)


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