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.