Further Investigation

In the last blog entry I suggested that a simple relationship governs the relationship of irreducible to reducible fractions (relating to the fractions in the Type 2 aspect of the number system that express the n roots of 1).

So again for example if n = 9, the n roots can be expressed in Type 2 terms as 1^1/9, 1^2/9, 1^3/9, 1^4/9, 1^5/9, 1^6/9, 1^7/9, 1^8/9 and 1^9/9.

So the nine fractions in question here are 1/9, 2/9, 3/9, 4/9, 5/9, 6/9, 7/9, 8/9 and 9/9.

Of these 1/9, 2/9, 4/9, 5/9, 7/9 and 8/9 are irreducible (as numerator and denominator have no common factors).

By contrast however 3/9, 6/9 and 9/9 are reducible!

So the hypothesis I offered was that the average proportion of irreducible fractions with respect to the number system as a whole → 1/(1 + 2/π) = π/(π + 2).

Therefore the average proportion of reducible fractions for the number system as a whole → 1/(1 + π/2) = 2/(π + 2).

This would entail that on average about 61.1% of fractions would be irreducible and 38.9% reducible.

Expressed even more simple the average ratio of irreducible to reducible fractions  → π/2 or alternatively the average ratio of reducible to irreducible fractions → 2/π. 

I then went on to consider the proportion of irreducible factors that would apply to the roots of those numbers with non-repeating and repeating prime structures respectively.

So excluding 1, the numbers between 2 and 10 with non-repeating prime structures are 2, 3, 5, 6, 7 and 10, whereas 4, 8 and 9 have repeating prime structures (relating to its constituent prime factors).

And again for the number system as a whole, the average proportion of numbers with non-repeating prime structures number → 1/(1 + 2/π) = π/(π + 2). And the corresponding proportion of numbers with repeating prime structures  → 1/(1 + π/2) = 2/(π + 2).  

Now one would expect that a higher proportion of irreducible factors would apply with respect to those numbers with non-repeating prime structures.

From my preliminary estimates it seems that another simple pattern comes into focus.

It would appear that with respect to the numbers with non-repeating prime structures that the average proportion of irreducible factors  → (π – 1)π = .68169....

This therefore would imply that the average proportion of irreducible factors for numbers with repeating prime structures  →.5.

Expressed another way, this would thereby imply that for  numbers with repeating prime structures the average proportion of both reducible and irreducible fractions would approach equality or alternatively that the ratio of reducible to the ratio of non-reducible factors (for  numbers with repeating prime structures) → 1.