A0 = dr(N) "A General Theory for N = pA + qB with p ≡ 1 (mod q)"

Key Points

• The aim is to establish a comprehensive theory for linear Diophantine systems where p and q are coprime and p ≡ 1 (mod q).
• Developed theoretical framework for linear Diophantine systems N = pA + qB.
• Established relationships involving the generalized digital root dr_q(N).
• Derived closed formula for representation count R(p,q;N) and examined arithmetic progressions.
• Identified that the minimal coefficient A₀ is equal to the generalized digital root dr_q(N).
• Provided a closed formula for the representation count R(p,q;N) with specific conditions.
• Demonstrated properties of arithmetic progressions in the context of these linear equations.

Abstract