Relative prime theory

Take some N relative primes defined by:

 sum of their squares equals N * their product. Since the relative primes will have fractions of mod N, then we know the number of fractional combinations is N/(N-1) ^ the product.

But we can then use Hurwitz theorem to prove there is an finite error for which the ratio maximizes an irrational approximation.   Thge maximum entrop extant, an Avogrado'1 number, is fixed per any inter number of relatice primes, N. Beyond that we must use N+1 primes.

In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξthere are infinitely many relatively prime integers m, n such that

${\displaystyle \left|\xi -{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}.}$

Example: (3/2)^(108) = (Phi)^91 to some very small error.

108 = 2^2 * 3^3, which corresponds to Shannon's entropic extend for three dimensional system.