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:
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\(\begin{matrix} \text{p}_1 = 2, & \text{p}_2 = 3, & \text{p}_3 = 5, & \text{p}_4 = 7, & \text{p}_5 = 11, & \text{p}_6 = 13, & \text{p}_7 = 17, & \text{p}_8 = 19, & \ldots \end{matrix}\) |
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:
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\(\begin{matrix} 9876543210 & = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1. \end{matrix}\) |
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:
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\(\begin{array}{rcl} 2 & \mapsto & \text{p}_1^1 \\[6pt] 3 & \mapsto & \text{p}_2^1 \\[6pt] 7 & \mapsto & \text{p}_4^1 \\[6pt] 32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1 \end{array}\) |
This leads to the following development:
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\(\begin{array}{lll} 9876543210 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1}^1 \text{p}_{\text{p}_4^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1}^1 \end{array}\) |
Continuing to replace every index and exponent with its factorization until no index or exponent remains unfactored produces the following development:
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\(\begin{array}{lll} 9876543210 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1}^1 \text{p}_{\text{p}_4^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^2}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^2}^1 \text{p}_{\text{p}_1^1 \text{p}_3^1}^1 \text{p}_{\text{p}_1^4}^1}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_2^1}^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^2}}^1}^1 \\[18pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1}^1 \end{array}\) |
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:
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\(\begin{array}{lll} 9876543210 & = & \text{p} \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}_{\text{p}}} \text{p}_{\text{p}_{\text{p}^{\text{p}}}}^{\text{p}} \text{p}_{\text{p}_{\text{p}} \text{p}_{\text{p}^{\text{p}}} \text{p}_{\text{p} \text{p}_{\text{p}_{\text{p}}}} \text{p}_{\text{p}^{\text{p}^{\text{p}}}}} \end{array}\) |
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.