Nested Prime Factorization (NPF)

[vak]

Появилась новая версия Грока, ну я её сразу и озадачил.

> Please propose a new revolutionary method of representing integer numbers.

The "Nested Prime Factorization" (NPF) system is a revolutionary method for representing integers that transforms the traditional prime factorization into a recursive, nested structure. This approach not only highlights the multiplicative nature of numbers but also creates a tree-like representation that is particularly suited for modern computational paradigms, such as tree-based machine learning models, recursive algorithms, and efficient storage of large numbers in hierarchical data structures. Unlike conventional positional systems (e.g., decimal or binary), which focus on additive place values, NPF emphasizes multiplication and recursion, making it easier to visualize and manipulate number theory properties like primality, exponents, and factor relationships. It is compact for highly composite numbers or high powers, reduces repetition in representation, and enables novel operations like "unfolding" the tree for factorization or "folding" for synthesis.

How the NPF System Works

1. Base Cases:

• 0 is represented as ∅ (empty set, symbolizing absence).
• 1 is represented as [1] (a singleton leaf).
• Negative integers are prefixed with a minus sign (-) applied to the representation of their absolute value (e.g., -n is -[representation of n]).

2. Positive Integers (n > 1):

• Start with the canonical prime factorization of n = p1^α1 * p2^α2 * ... * pk^αk, where p1 < p2 < ... < pk are primes and αi > 0.
• Represent n as a list [R(α1), R(α2), ..., R(αk)], where R(αi) is the recursive representation of the exponent αi.
• Recurse on each exponent αi until reaching base cases (1 or primes, which are leaves).
• Primes are represented as themselves in leaves, but if an exponent is prime, it is left as is unless further decomposed (primes stop recursion).

3. Recursion Rule:

• If an exponent α is 1, it is [1].
• If α is prime, it is the prime value (e.g., 2, 3, 5).
• If α is composite, apply the NPF recursively to α.

This creates a nested list structure, which can be serialized in text using brackets or JSON-like notation for clarity. The structure is unique for each n, ensuring bijectivity.

Why Revolutionary?

• Hierarchical Insight: The nested structure mirrors the "building blocks" of numbers, making it intuitive for educational purposes, number theory analysis, and AI applications (e.g., input to graph neural networks for predicting properties like divisibility).
• Compactness for Large Numbers: High exponents are nested deeply rather than repeated, reducing representation length compared to linear factorization strings. For example, 2^{64} would be a deeply nested [ [...] ] rather than "2^64".
• Operational Efficiency: Arithmetic can be performed recursively—multiplication concatenates or nests lists, addition requires "merging" trees with carries resolved via recursion. This could accelerate computations in symbolic math software or quantum simulation, where nested structures align with qubit entanglement models.
• Extension to Other Domains: Easily adapts to rationals (numerator/denominator as two trees) or complexes, and supports error-resilient encoding by adding redundancy in nests.

Examples

• 0: ∅
• 1: [1]
• -1: -[1]
• 2 (prime): 2
• 3 (prime): 3
• 4 = 2^2: [2] (since exponent 2 is prime)
• -4: -[2]
• 6 = 2^1 * 3^1: [ [1], [1] ]
• 8 = 2^3: [3] (exponent 3 is prime)
• 12 = 2^2 * 3^1: [2, [1]]
• 16 = 2^4 = 2^(2^2): [[2]] (recursive on exponent 4 = 2^2)
• 1200 = 2^4 * 3^1 * 5^2: [ [2], [1], 2 ] (exponent 4 = [2], 1 = [1], 2 = 2)
• 65536 = 2^16 = 2^(2^4): [[[2]]] (deep nesting for 16 = 2^4, 4 = 2^2, 2 = 2)
• -65536: -[[[2]]]

Comparison with Traditional Systems

The table below compares NPF with binary, decimal, and factorial representations for select numbers, highlighting length (characters in serialized form) and key advantages.

| Number | Binary      | Decimal | Factorial     | NPF (Lists) | Notes on NPF Advantage 
|--------|-------------|---------|---------------|-------------|------------------------
| 1      | 1           | 1       | 1!0           | [1]         | Minimal nesting for unity.
| 4      | 100         | 4       | 2!0           | [2]         | Reveals prime exponent directly.
| 8      | 1000        | 8       | 2!1!0         | [3]         | Compact for power-of-2. 
| 12     | 1100        | 12      | 2!2!0         | [2,[1]]     | Shows factors separately.
| 1200   | 10010110000 | 1200    | 4!3!2!1!0!0   | [[2],[1],2] | Nested for exponents, shorter than linear factorization (2^4*3^1*5^2 = 15 chars).
| 65536  | (17 bits)   | 65536   | Long sequence | [[[2]]]     | Highly compact for exponential towers; binary requires 17 chars, NPF uses 7.

This system opens doors to new computational techniques, such as recursive traversal for fast GCD (by comparing nests) or visualization as trees for educational tools. Future extensions could include probabilistic nesting for approximate representations in AI training data.