The Fundamental Theorem of Arithmetic - Part 2

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

I left off with the following

The Fundamental Theorem of Arithmetic is an infinite product of prime geometric power series.

$$\left(\begin{array}{c}\\ 2^n\\ +\\ ⋮\\ 2^5\\ +\\ 2^4\\ +\\ 2^3\\ +\\ 2^2\\ +\\ 2^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 3^n\\ +\\ ⋮\\ 3^5\\ +\\ 3^4\\ +\\ 3^3\\ +\\ 3^2\\ +\\ 3^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 5^n\\ +\\ ⋮\\ 5^5\\ +\\ 5^4\\ +\\ 5^3\\ +\\ 5^2\\ +\\ 5^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 7^n\\ +\\ ⋮\\ 7^5\\ +\\ 7^4\\ +\\ 7^3\\ +\\ 7^2\\ +\\ 7^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {11}^n\\ +\\ ⋮\\ {11}^5\\ +\\ {11}^4\\ +\\ {11}^3\\ +\\ {11}^2\\ +\\ {11}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {13}^n\\ +\\ ⋮\\ {13}^5\\ +\\ {13}^4\\ +\\ {13}^3\\ +\\ {13}^2\\ +\\ {13}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {17}^n\\ +\\ ⋮\\ {17}^5\\ +\\ {17}^4\\ +\\ {17}^3\\ +\\ {17}^2\\ +\\ {17}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {19}^n\\ +\\ ⋮\\ {19}^5\\ +\\ {19}^4\\ +\\ {19}^3\\ +\\ {19}^2\\ +\\ {19}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {23}^n\\ +\\ ⋮\\ {23}^5\\ +\\ {23}^4\\ +\\ {23}^3\\ +\\ {23}^2\\ +\\ {23}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {29}^n\\ +\\ ⋮\\ {29}^5\\ +\\ {29}^4\\ +\\ {29}^3\\ +\\ {29}^2\\ +\\ {29}^1\\ +\\ 1\\ \end{array}\right)\times \cdots$$

There are two models we can use to examine factorization or multiplicative distribution of the above infinite product of prime geometric power series.

The first model is just a visual model of tracing distinct/unique multiplicative paths through the above infinite product of prime geometric power series.

The second model or method is to create a countably infinite multiplicative sequence.

Rules for the multiplicative sequences modeling the multiplicative distribution of the infinite product of prime geometric power series.

• A multiplicative sequence must be distinct (unique).
  
  
• A multiplicative sequence must have infinite terms, as there is a one-to-one correlation between a prime power series term and a term in the countably infinite multiplicative sequence model. There are countably infinite prime power series terms in the infinite product. There will be countably infinite terms in a multiplicative sequence.
  
  
• The values that are selectable as a choice for each term are either 1 or an integer power of the respective prime power series.
  
  
• A value must be selected for each and every infinite term.

Listing multiplicative sequences in the order of the natural numbers.

Finitely factored sequences

$$\begin{array}{cl}n& \text{sequence}\\ 1& 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 2& 2^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 3& 1,3^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 4& 2^2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 5& 1,1,5^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 6& 2^1,3^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 7& 1,1,1,7^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 8& 2^3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 9& 1,3^2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 10& 2^1,1,5^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 11& 1,1,1,1,{11}^1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 12& 2^2,3^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 13& 1,1,1,1,1,{13}^1,1,1,1,1,1,1,1,1,1,1,1,1,...\end{array}$$

Listing multiplicative sequences can continue this way infinitely.

As we see from the above, the infinite product of prime geometric power series is basically 'The Fundamental Theorem of Arithmetic'.

In the next post, I will look at the inverse naturals and the harmonic series, thanks to a great mathematician, Leonhard Euler.

Continue to - The Inverse Naturals and The Euler Product