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Instead of looking at the number of primes below X, we fix the number of primes P (say 10 or 100...) and then we look at how the length of the successive intervals having the same fixed number of primes P $$(a_{1},b_{1}),(a_{2},b_{2}),...(a_{n},b_{n})$$ changes.

Here, $$a_{i},b_{i}$$ are composite, first and last.

I tried to see if we can get an idea by using a table of the first 10,000 primes but it turned out that the sample is too small. But it showed that P should be much larger than 10 or 100 because of the fluctuations of the prime numbers. A much bigger number P will not be affected by these fluctuations. Unfortunately, I do not code so I won't be able to conduct these numerical experiments.

Can we find an approximate formula for these intervals? Suppose we find an exact closed form, will it not be a way to determine the next prime?

user25406
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    Note that if $P=2$ we would be asking about the sizes of prime gaps, a fairly important topic. – hardmath Feb 03 '17 at 17:11
  • I am pretty sure that an expression ( formula that most probably involves natural logarithms) that describes the length of these intervals can be derived but I just don't know enough to do that. – user25406 Feb 03 '17 at 18:30
  • My point is that a lot of highly motivated folks, smarter than I am, have looked at the distribution of prime gaps without narrowing the distance between upper limits and lower limits much. The twin prime conjecture says that gaps of two will occur infinitely often, and this is still an open problem. However it is known that there is some modest size prime gap that occurs infinitely often thanks to the work of Zhang and those who improved on his breakthrough. – hardmath Feb 03 '17 at 19:09

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