The prime factorization numbering system

 Introduction:

Within the realm of mathematical systems and numerical representation, I have invented a method to represent numbers using their prime factorization. This invention is none other than the Prime Factorization Numbering System, a creation born from a comprehensive study of various numerical systems. In this page, we will explore how this system functions, its practical applications, and its potential for expansion by incorporating additional prime numbers into the symbol set.

The representation of the basic primes:

At the core of this system lies the utilization of prime numbers as fundamental building blocks, each endowed with a distinctive symbol:

–1 is represented as 'U'

–2 is represented as 'B'

–3 as 'T'

–5 as 'P'

–7 (dec) as 'S'

–11 (dec) as 'L'

–13 (dec) as 'D'

These prime symbols form the basis of the system's numerical representation, offering an efficient means to represent 13-smooth numbers.

 

Representation of Composite Numbers and Larger Primes:

One of the system's key features is its multiplicative nature, setting it apart from traditional numerical systems. When two numbers are multiplied, their prime factorizations combine, yielding compact representations. For instance, 6 (dec) is represented as 'BT' since 6 (dec) = 2 × 3, and 77 (dec) is represented as 'SL' because 77 (dec) = 7 (dec) × 11 (dec). It's important to note that the 'U' symbol cannot be used for multiplications since every number should have a unique representation. Additionally, prime factors must be presented in the correct order to ensure that each 13-smooth number has exactly six different representations. To represent zero, use the symbol '-' to indicate null.

For larger prime numbers, such as 17 (dec) and 119594217887 (dec), subtract one from the number and calculate the representation of the preceding number. For example, 17 (dec) is represented as U(BBBB), and 119594217887 (dec) is represented as U(BLLLLLDDDDD) since it is a prime number. In some cases, primes may require more than one 'U', as seen in 2879 (dec), represented as U(BU(BU(BU(BU(BU(BBBL)))))).

I apologize for the oversight. Including examples of exponentiation in action is indeed important to illustrate the practical application of this feature within your Prime Factorization Numbering System. Here's an updated section with examples of exponentiation:

Representation of Negatives, Rationals, and Exponentiation:

To represent negative numbers, append the 'N' symbol to indicate a negative value. For example, -1573 (dec) is represented as NLD. For rational numbers, use the 'R' symbol to represent the '/' symbol. For instance, 3/2 is represented as TRB, and -5/3 is NPRT.

Sometimes, numbers can become quite large if they possess numerous factors of a specific prime. In such cases, the 'E' symbol can be employed to denote exponentiation, reducing the number of symbols required. The 'E' symbol can be used for any prime, with the condition that the numbers following the 'E' symbol must be positive integers, precluding the use of rational or negative numbers. Exponentiation can also be stacked, allowing for the representation of exceptionally large numbers.

Here are more examples of how exponentiation simplifies representations:

–65537 (dec) can be represented as U(BBBBBBBBBBBBBBBB), but with the 'E' symbol, it can be represented as U(BE(BBBB)). This demonstrates how exponentiation streamlines the representation of the same number.

–49013608176639571426873298708473317965912267 (dec) can be represented as U(BSSSSSSSSSSSSSSSSSSSSSSSLLLLLLLLLLLLLLLLLLLLLLL), but it can also be represented as U(BSE(U(BL))LE(U(BL))), showcasing the efficiency gained through exponentiation.

This feature highlights the versatility of the Prime Factorization Numbering System in efficiently representing numbers of considerable magnitude while maintaining a concise and logical notation.

Expansions and Utility of the System:

To enhance efficiency, consider inventing symbols for prime numbers beyond 13 (dec) while ensuring that the symbols maintain an orderly progression. This expansion will simplify the representation of a wider range of numbers and reduce the number of characters needed.

The Prime Factorization Numbering System serves as a neutral numerical representation system, valuing numbers based on their factorizations rather than arbitrary values. This system is particularly useful for exact number representation and as a neutral base for positional numbering systems. However, it is not well-suited for addition or subtraction operations and is primarily a representative system rather than a system for calculations.

Conclusion:

The Prime Factorization Numbering System offers a neutral and efficient means of representing numbers. Its adaptability is evident in its ability to efficiently represent numbers, large primes, negative numbers, and rational numbers (except zero). Furthermore, the introduction of exponentiation allows for the representation of extremely large numbers. It can very easily replace the Roman numerals, which are decimal based, since they share the fact that both systems represent exact quantities, and not approximations. As this system continues to evolve, it may become even more efficient for representing larger numbers and different types of numbers, such as algebraic or complex numbers.