Another Interesting Relationship

In an earlier blog entry, “Remarkable Features of the Number System 1”, I drew attention to a very simple relationship governing the ratio of numbers with non-repeating to numbers with repeating prime structures respectively.

Once again when each prime occurs but once in the unique factor composition of a number, then it is termed as a number with a non-repeating structure.

So for example 30 = 2 * 3 * 5 represents a number with a non-repeating prime structure (as each factor occurs but once).

However when one or more primes is repeated in the unique factor composition of that number then it is a number with a repeating prime structure.

So in this context, for example 28 = 2 * 2 * 7 represents a number with a repeating prime structure (as 2 in this case occurs twice).

Basically, I concluded following a fairly extensive range of empirical testing, that for the number system as a whole, the average frequency of numbers with non-repeating prime structures → 1/(1 + 2/π)  = π/(π + 2) and that the corresponding average frequency of numbers with repeating prime structures → 1/(1 + π/2) = 2/(π  + 2).

Therefore the ratio of numbers with non-repeating to repeating prime structures (for the number system as a whole) → π/2.

Alternatively, we could say that the ratio of numbers with repeating to non-repeating prime structures (for the number system as a whole) → 2/π.   

Now of course this represents a Type 1 view of number where the unique prime factors of each number is expressed with respect to the default dimensional power of 1.

So 3 for example as a constituent factor, is more fully expressed = 3¹.

Recently my attention turned to what in fact represents a complementary type problem.

We can view the various roots of a number in Type 2 terms, where now in inverse terms, the default base number of 1 is raised to dimensional powers that can vary.

So for example the 3 roots of 1 would thereby be expressed as 1^1/3, 1^2/3 and 1^3/3 respectively. Thus concentrating on the dimensional values (representing the Type 2 notion of number) the three values are 1/3, 2/3 and 3/3 respectively.

In more general terms, the n roots of 1 - again concentrating on the dimensional values - will range over all the natural numbers from 1/n to n/n.

Now clearly where these reflect the prime roots of 1, when we exclude the final fractional value (which always reduces to 1) all the other fractional values will be irreducible. In other words it will not be possible to reduce any of these factors to a smaller fraction (as no common factor can exist with respect to both numerator and denominator).

However where a composite number n is involved, the n roots of 1 will then yield fractional values where some are reducible and others non-reducible.

For example if we take the 12 roots of 1 (where of course 12 is composite) the 12 fractional values generated will be 1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 8/12, 9/12, 10/12, 11/12 and 12/12.

Now, we can easily see that 1/12, 5/12, 7/12 and 11/12 are irreducible fractions.

However the remaining fractions here i.e. 2/12, 3/12, 4/12, 6/12, 8/12, 9/12, 10/12 and 12/12 are reducible (with both numerator and denominator containing common factors). 

An interesting question then arises with respect to the number system as a whole, as to the average frequency of fractional values that are irreducible and reducible respectively.

Remarkably, this parallels closely the earlier relationship as to the average frequency of numbers with non-repeating and repeating prime structures respectively.

So the hypothesis that I now offer is that the average frequency of fractional values that are irreducible → 1/(1 + 2/π)  = π/(π + 2); then the average frequency of fractional values that are reducible → 1/(1 + π/2) = 2/(π + 2).

So, therefore the ratio of irreducible to reducible fractions → π/2.

Alternatively, the ratio of reducible to irreducible fractions → 2/π.

Now, I counted the irreducible fractions for roots of all numbers to 100 = 3054 (approx). relative to all fractions (5050). This works out at .60475 which is slightly less than π/(π + 2) = .61101.

Now in counting up irreducible fractions, the primes make the greatest contribution. So if n is a prime n – 1 will be irreducible fractions.

The formula n (log n – 1) predicts 47 primes up to 200 with the actual occurrence = 46.

However it predicts 28 up to 100 (where the actual occurrence = 25.

This would suggest that the actual frequency of primes is less than what would be generally expected up to 100 which accounts in large measure for the underestimate that I obtained.

However if one counts all fractions to 110 where the actual no. of primes = 29 against a predicted value of 30, one now gets the much better estimate of 3726/6105 = .6103.

So there is little doubt to my mind that the formula I have suggested is the correct one, bearing a direct complementary (Type 2) relationship to the earlier (Type 1) that was mentioned in relation to the average frequency of numbers with non-repeating primes.

In fact the intuitive realisation of this fact had already suggested to me what the answer would be before I actually carried out any numerical calculations to verify its nature.