Linear Diophantine Representation Systems — p ≡ 1 (mod q) provides an extensive theoretical framework for understanding linear Diophantine systems expressed as N = pA + qB under the condition that p is congruent to 1 modulo q. This project reveals that the minimal A-value correlates directly to the digital root of N, offering foundational insights in number theory and its applications.

Core Results

• Main Theorem: The relationship A₀ = dr(N) holds for every N that is greater than or equal to pq.
• Representation Formula: The representation formula is defined as R(N) = ⌊(⌊N/p⌋ − dr(N)) / q⌋ + 1, showcasing how digital roots interact with these systems.
• Family Boundaries: The boundaries are expressed by the formula G(k) = 171k − 27.
• Additivity Theory: This principle states that if a + b = c, then the representations of the numbers sum simultaneously across three distinct levels: numeric values, coordinates, and digital roots.
• Matryoshka Theory: It posits that every output in the reduction chain N → S(N) → dr(N) can be represented as an A-coordinate in N's representations.

Additionally, this project includes proof that (19, 9) forms the unique Class-IV pair within this theoretical framework.

Implementation

Python implementations are provided for further exploration and experimentation, available in the files core.py, system_19_9.py, and matryoshka.py.

For further academic insights, refer to the publication here. This work significantly contributes to the understanding of linear Diophantine equations and their underlying principles.