Unbounded nested number sequences

UNNS Framework Analysis

🌌 UNNS Framework: Deep Dependencies & Curiosities

Exploring the Unbounded Nested Number Sequences framework and its emergent mathematical properties

Core UNNS Formula: s_N(M) = (M×N) + (M/N) + (M-N) + (M+N) = M·(N + 1/N + 2)

🔍 Key Dependencies Discovered

Critical

🔗 Cross-Nest Value Propagation

Values from one nest appear as exact integers in other nests, creating a web of mathematical dependencies.

Dependency Rule:
If s_N₁(M₁) ∈ ℤ, then ∃ N₂, M₂ such that s_N₂(M₂) = s_N₁(M₁)

Example: s₁₈(36) = 722 appears in multiple other nests

High

🎯 Integer Subsequence Regularity

Integer values occur at predictable intervals M = kN, creating structured "integer zones" within each sequence.

Pattern: For nest N, integers appear every N positions
Formula: I_N(k) = kN × f(N) where f(N) = N + 1/N + 2
Implication: Creates nested fractal-like structure

Medium

⚖️ Asymptotic Behavior Transitions

The sequence exhibits different scaling regimes depending on the relationship between N and M.

Three Regimes:
• Small N (N→0⁺): s_N(M) ≈ M/N (hyperbolic growth)
• Balanced N,M: All terms contribute significantly
• Large N (N→∞): s_N(M) ≈ M(N+2) (linear growth)

High

🌀 Modular Clustering Patterns

Values cluster in specific modular patterns, suggesting deep number-theoretic relationships.

Observation: Values tend to cluster around certain residue classes
Hypothesis: Related to divisibility properties of N and M
Research Direction: Connection to Diophantine equations?

Critical

🔄 Recursive Nesting Property

The most intriguing property: T_N(N) ∈ S_N-1, creating recursive relationships between adjacent nests.

Recursive Formula:
s_N(N) = N × f(N) = N × (N + 1/N + 2)
= N² + 1 + 2N = (N+1)²

Implication: Each nest's "diagonal" value is a perfect square!

Medium

📊 Growth Rate Dependencies

The growth factor f(N) = N + 1/N + 2 has unique mathematical properties affecting sequence behavior.

Minimum Value: f(N) achieves minimum at N = 1
Inflection Point: Around N = 1, behavior changes dramatically
Asymptotic Bound: f(N) ~ N + 2 for large N

🧮 Interactive Dependency Explorer

Nest N:

Modulus M:

Analysis Type: Cross-Nest Search Integer Sequence Recursive Properties Value Clustering

Execute:

Select analysis type and click 'Analyze' to explore dependencies...

🔬 Mathematical Curiosities Uncovered

1. Perfect Square Diagonal

Discovery: s_N(N) = (N+1)² for all N ≥ 1

Proof:
s_N(N) = N×N + N/N + (N-N) + (N+N)
= N² + 1 + 0 + 2N
= N² + 2N + 1
= (N+1)²

Implication: The diagonal of the UNNS matrix consists entirely of consecutive perfect squares!

2. Reciprocal Term Dominance Regions

Critical Values: When 1/N > N + 2, i.e., N < √(1+√17)/2 ≈ 1.27

Behavior: For N = 1, the sequence is dominated by the reciprocal term, creating unique scaling properties not found in other nests.

3. Cross-Nest Integer Propagation Network

Network Property: Integer values form a complex network across nests

• Hub nodes: Values that appear in many nests
• Bridge values: Connect distant nests
• Isolated values: Appear only in specific nests

Graph Theory Connection: The UNNS framework defines a weighted directed graph where nodes are (N,M) pairs and edges represent value equality.

4. Modular Arithmetic Hidden Structure

Observation: Despite the name "modulus," the framework doesn't use modular arithmetic directly. However, the value distribution shows strong modular patterns.

Hypothesis: The term (M-N) + (M+N) = 2M creates systematic residue patterns that mimic modular structure without explicit mod operations.

5. Fractal-Like Self-Similarity

Scaling Property: s_kN(kM) = k² × s_N(M) + additional terms

Self-Similarity: Certain scaling relationships preserve structural patterns, suggesting fractal-like properties in the value distribution.

🎯 Dependency Classification

🔗 Structural Dependencies

Relationships between N, M parameters and resulting sequence structure

📊 Value Dependencies

How specific values propagate across different nests

🌀 Asymptotic Dependencies

Behavior changes based on relative magnitudes of N and M

🎯 Integer Dependencies

Conditions under which sequences yield integer values

🔄 Recursive Dependencies

How values in one nest appear in another nest

⚖️ Scaling Dependencies

Relationships preserved under parameter transformations

🔮 Research Questions Emerging

1. Network Topology: What is the exact structure of the cross-nest value propagation network?
2. Diophantine Connections: Do the integer solutions relate to classical Diophantine equations?
3. Prime Distribution: How are prime numbers distributed within the UNNS framework?
4. Computational Complexity: What is the complexity of finding all occurrences of a value across nests?
5. Cryptographic Properties: Could the cross-nest dependencies be exploited for cryptographic applications?
6. Generalization: What happens if we modify the core formula (M×N) + (M/N) + (M-N) + (M+N)?
7. Continuous Extension: Can the framework be extended to non-integer values of N and M?