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April 26, 20260 citationsOpen Access

A General Theory for N = pA + qB with p ≡ 1 (mod q)

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BIBilal El Issaoui

Key Points

  • To establish a complete theory for the linear Diophantine systems N = pA + qB with specific properties of p and q.
  • Develops a mathematical framework for coprime pairs (p, q) where p ≡ 1 (mod q).
  • Derives a closed formula for the minimal coefficient A₀ and representation count R(p, q; N).
  • Explores arithmetic progressions within the defined systems.
  • The minimal coefficient A₀ equals the generalized digital root dr_q(N).
  • A closed formula for the representation count R(p, q; N) is presented.
  • The findings reveal patterns in the arithmetic progressions connected to the systems.

Abstract

This paper develops a complete theory of linear Diophantine systems N = pA + qB for coprime pairs (p, q) satisfying p ≡ 1 (mod q). The central result establishes that the minimal coefficient A₀ equals the generalised digital root drq (N), with a closed formula for the representation count R (p, q;N) and arithmetic progressions.

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Cite This Study

Bilal El Issaoui (2026) studied this question.

synapsesocial.com/papers/69edacdb4a46254e215b4a01https://doi.org/10.5281/zenodo.19719783
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