# Directory talk:Jon Awbrey/Papers/Riffs and Rotes

## Place for Discussion

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## Idea (Old Version)

Let $$\text{p}_i$$ be the $$i^\text{th}$$ prime, where the positive integer $$i$$ is called the index of the prime $$\text{p}_i$$ and the indices are taken in such a way that $$\text{p}_1 = 2.$$ Thus the sequence of primes begins as follows:

 $$\begin{matrix} \text{p}_1 = 2, & \text{p}_2 = 3, & \text{p}_3 = 5, & \text{p}_4 = 7, & \text{p}_5 = 11, & \text{p}_6 = 13, & \text{p}_7 = 17, & \text{p}_8 = 19, & \ldots \end{matrix}$$

The prime factorization of a positive integer $$n$$ can be written in the following form:

 $$n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},$$

where $$\text{p}_{i(k)}^{j(k)}$$ is the $$k^\text{th}$$ prime power in the factorization and $$\ell$$ is the number of distinct prime factors dividing $$n.$$ The factorization of $$1$$ is defined as $$1$$ in accord with the convention that an empty product is equal to $$1.$$

Let $$I(n)$$ be the set of indices of primes that divide $$n$$ and let $$j(i, n)$$ be the number of times that $$\text{p}_i$$ divides $$n.$$ Then the prime factorization of $$n$$ can be written in the following alternative form:

 $$n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.$$

For example:

 $$\begin{matrix} 9876543210 & = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1. \end{matrix}$$

Each index $$i$$ and exponent $$j$$ appearing in the prime factorization of a positive integer $$n$$ is itself a positive integer, and thus has a prime factorization of its own.

Continuing with the same example, the index $$32277$$ has the factorization $$3 \cdot 7 \cdot 29 \cdot 53 = \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1.$$ Taking this information together with previously known factorizations allows the following replacements to be made:

 $$\begin{array}{rcl} 2 & \mapsto & \text{p}_1^1 \\[6pt] 3 & \mapsto & \text{p}_2^1 \\[6pt] 7 & \mapsto & \text{p}_4^1 \\[6pt] 32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1 \end{array}$$

This leads to the following development:

 $$\begin{array}{lll} 9876543210 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1}^1 \text{p}_{\text{p}_4^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1}^1 \end{array}$$

Continuing to replace every index and exponent with its factorization until no index or exponent remains unfactored produces the following development:

 $$\begin{array}{lll} 9876543210 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1}^1 \text{p}_{\text{p}_4^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^2}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^2}^1 \text{p}_{\text{p}_1^1 \text{p}_3^1}^1 \text{p}_{\text{p}_1^4}^1}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_2^1}^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^2}}^1}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1}^1 \end{array}$$

The $$1$$'s that appear as indices and exponents are formally redundant, conveying no information apart from the places they occupy in the resulting syntactic structure. Leaving them tacit produces the following expression:

 $$\begin{array}{lll} 9876543210 & = & \text{p} \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}_{\text{p}}} \text{p}_{\text{p}_{\text{p}^{\text{p}}}}^{\text{p}} \text{p}_{\text{p}_{\text{p}} \text{p}_{\text{p}^{\text{p}}} \text{p}_{\text{p} \text{p}_{\text{p}_{\text{p}}}} \text{p}_{\text{p}^{\text{p}^{\text{p}}}}} \end{array}$$

An expression of this form may be referred to as the doubly recursive factorization (DRF) or drift of the positive integer from which it derives.