This prime number theory

The number of primes will be Nth prime will be nLog(n) where n is the nuber of primes.

1^1, 2^23, 3^3.

If you had a fair coin, what is the probability of getting 5 heads in a row?  That is equivalent to finding a sequence with a path through each of N primes.  The point of maximum entropy where you have the greatest number of combinations to using n primes in a sequence..

That is the top of the binomial, and the largest term in the exponential series.  It should have the greatest round off bandwidth and so make the best approximation to the irrational. Number paths go by factorial of the primes. But take the prime sequence as a Markov n-tuple, and you see that you have (n-1)/n spare paths, and still meet the condition. That tends to one. We are making the round off longer and more accurate.

But no amount of primes will count to the most irrational round off space. So each each n-tuple configuration has a maximum approximation, it is done and would otherwise repeat. Simple enough, add 1 to the number of primes and bump to the next n-tuple, at the node with the proper N count.

Markov tuple are relatively prime, and increasing N should make them relatively prime and more dense in the prime Bayesian space.

So we have this series of multiple dimensional Markov trees and we want to predict when the node of an N+1 tree is close to the node if a N tree. The gaps are the sparsity of one tree relative to the other, and one is a compression of the other.

Sample space rules apply. If I have 2/3 duplicates, my sample base is{
 3/2 = 2-1/2.
The next base is 2-2/3. And so one.  My exponent on this is my maximum, which should be prime to the prime power. I can convert that base to something like 2^n * (1-(n-1/n))^n.  The right term is the binomial with the unfair coin.   I can easily expand from N to N+1 because I am counting my  Avogadro by one will be, the i^i series, over the number of relative primes.   Just work those two equations.

Crossing Markov trees gets you the optimum set of relative primes for the current Avogadro. But it gives you the best approximation to the primes given some N in its range.