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Regarding the article below, it was mentioned that prime numbers can only end with digits 1,3,7 or 9 (except for 2 and 5) but that the frequency of that these digits occur are not random, as can be seen from the fact that it is unlikely that the same digit will occur twice for consecutive prime numbers. However the article also mentions that as the prime number tends to infinity, then the distribution of prime numbers also tends to become random. If it is true then how do we prove this fact and can we conclude that the distribution of primes is random or whether it is not random and there is in fact a pattern to it?

http://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html

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    $\begingroup$ By definition the prime number distribution can not be "random". But I'll cede the point for lack/difficulty of a better term. The pattern becomes more "random" for sequences of larger primes but that still means the earlier distribution is has the property and are not "random". No one at this time knows how to prove this result or why it is true. $\endgroup$ – fleablood Aug 5 '16 at 19:14
  • $\begingroup$ @fleablood that bothers me too, it is really more accurate to talk about statistics of the primes in this way. The digits mentioned are statistically less likely to repeat earlier on. $\endgroup$ – Carser Aug 5 '16 at 19:19
  • $\begingroup$ What i think is that if we consider a small subset of primes then we may find that those primes does not appear randomly but say if we consider infinitely many primes then we can in fact see that the primes does occur randomly that means that each of the digits 2,3,7 ,9 has a 0.25 percent chance of occuring.? $\endgroup$ – ys wong Aug 5 '16 at 19:27
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    $\begingroup$ It gets closer to .25. But even if there comes a point where it becomes .25 and from that point on the primes are "random", that doesn't change the fact that this was significant for primes less than that point. $\endgroup$ – fleablood Aug 5 '16 at 20:10
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    $\begingroup$ "if we consider a small subset of primes" By no stretch of the imagination can 400 billion be considered a "small subset" and with a variation of 6.8 off the projected 25% and with a steady and observable increase from 18.2 to almost 25% than something is clearly going on. And don't forget that as primes get larger they become less common. There is clearly so "non 'random' conspiracy" going on, the strength of which is related to the density (how common) primes are. Even if it dwindles to zero, it is still there. $\endgroup$ – fleablood Aug 5 '16 at 20:19

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