Changes

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<math>\begin{matrix}

<math>\begin{matrix}

& = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721

& = & 3^2 \cdot 3607 \cdot 3803

& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1.

& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1.

\end{matrix}</math>

\end{matrix}</math>

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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.

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.

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:

Continuing with the same example, the index <math>504</math> has the factorization <math>2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1</math> and the index <math>529</math> has the factorization <math>{23}^2 = \text{p}_9^2.</math>  Taking this information together with previously known factorizations allows the following replacements to be made in the above expression:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

2 & \mapsto & \text{p}_1^1

2 & \mapsto & \text{p}_1^1

\\[6pt]

\\[6pt]

3 & \mapsto & \text{p}_2^1

504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1

\\[6pt]

\\[6pt]

7 & \mapsto & \text{p}_4^1

529 & \mapsto & \text{p}_9^2

\end{array}</math>

\end{array}</math>

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1

& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1

\\[12pt]

\\[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

& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1

\end{array}</math>

\end{array}</math>

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

Continuing to replace every index and exponent with its factorization produces the following development:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1

& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1

& = & \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]

\\[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

& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1

\\[18pt]

\\[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

& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_2^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^2}^1}^1 \text{p}_{\text{p}_{\text{p}_2^2}^{\text{p}_1^1}}^1

\\[18pt]

\\[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

& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^{\text{p}_1^1}}^1

\end{array}</math>

\end{array}</math>

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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:

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

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

& = & \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}}}}}

& = & \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}</math>

\end{array}</math>

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