What question did this study set out to answer?

To establish a complete theory for the linear Diophantine systems N = pA + qB with specific properties of p and q.

April 26, 2026Open Access

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

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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Bilal El Issaoui (Fri,) studied this question.

synapsesocial.com/papers/69edacdb4a46254e215b4a01 — DOI: https://doi.org/10.5281/zenodo.19719783

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

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