https://mywikibiz.com/index.php?title=Directory_talk:Jon_Awbrey/Papers/Riffs_and_Rotes&feed=atom&action=historyDirectory talk:Jon Awbrey/Papers/Riffs and Rotes - Revision history2024-03-29T06:25:07ZRevision history for this page on the wikiMediaWiki 1.35.3https://mywikibiz.com/index.php?title=Directory_talk:Jon_Awbrey/Papers/Riffs_and_Rotes&diff=107956&oldid=prevJon Awbrey: move old intro to talk page2010-02-04T12:22:46Z<p>move old intro to talk page</p>
<p><b>New page</b></p><div>==Place for Discussion==<br />
<br />
<br><math>\cdots</math><br><br />
<br />
==Idea (Old Version)==<br />
<br />
Let <math>\text{p}_i</math> be the <math>i^\text{th}</math> prime, where the positive integer <math>i</math> is called the ''index'' of the prime <math>\text{p}_i</math> and the indices are taken in such a way that <math>\text{p}_1 = 2.</math> Thus the sequence of primes begins as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\text{p}_1 = 2, &<br />
\text{p}_2 = 3, &<br />
\text{p}_3 = 5, &<br />
\text{p}_4 = 7, &<br />
\text{p}_5 = 11, &<br />
\text{p}_6 = 13, &<br />
\text{p}_7 = 17, &<br />
\text{p}_8 = 19, &<br />
\ldots<br />
\end{matrix}</math><br />
|}<br />
<br />
The prime factorization of a positive integer <math>n</math> can be written in the following form:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},</math><br />
|}<br />
<br />
where <math>\text{p}_{i(k)}^{j(k)}</math> is the <math>k^\text{th}</math> prime power in the factorization and <math>\ell</math> is the number of distinct prime factors dividing <math>n.</math> The factorization of <math>1</math> is defined as <math>1</math> in accord with the convention that an empty product is equal to <math>1.</math><br />
<br />
Let <math>I(n)</math> be the set of indices of primes that divide <math>n</math> and let <math>j(i, n)</math> be the number of times that <math>\text{p}_i</math> divides <math>n.</math> Then the prime factorization of <math>n</math> can be written in the following alternative form:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.</math><br />
|}<br />
<br />
For example:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
9876543210<br />
& = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721<br />
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1.<br />
\end{matrix}</math><br />
|}<br />
<br />
Each index <math>i</math> and exponent <math>j</math> appearing in the prime factorization of a positive integer <math>n</math> is itself a positive integer, and thus has a prime factorization of its own.<br />
<br />
Continuing with the same example, the index <math>32277</math> has the factorization <math>3 \cdot 7 \cdot 29 \cdot 53 = \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1.</math> Taking this information together with previously known factorizations allows the following replacements to be made:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{array}{rcl}<br />
2 & \mapsto & \text{p}_1^1<br />
\\[6pt]<br />
3 & \mapsto & \text{p}_2^1<br />
\\[6pt]<br />
7 & \mapsto & \text{p}_4^1<br />
\\[6pt]<br />
32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1<br />
\end{array}</math><br />
|}<br />
<br />
This leads to the following development:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
9876543210<br />
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1<br />
\\[12pt]<br />
& = & \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<br />
\end{array}</math><br />
|}<br />
<br />
Continuing to replace every index and exponent with its factorization until no index or exponent remains unfactored produces the following development:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
9876543210<br />
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1<br />
\\[18pt]<br />
& = & \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<br />
\\[18pt]<br />
& = & \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<br />
\\[18pt]<br />
& = & \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<br />
\\[18pt]<br />
& = & \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<br />
\end{array}</math><br />
|}<br />
<br />
The <math>1</math>'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:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
9876543210<br />
& = & \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}}}}}<br />
\end{array}</math><br />
|}<br />
<br />
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.</div>Jon Awbrey