Linear Equations with Fractions: The Frobenius Problem with Fractions

This work explores a structural approach to the Frobenius problem using linear equations of the form (N=19a+9b), where the relationship between the coefficients enables a direct remainder-based method for determining integer representations. It extends this approach to fractional values through scaling, allowing numbers to be expressed using fractional coordinates even when no non-negative integer representation exists. The framework also introduces a broader family of related systems based on (P=mQ+1), connecting modular arithmetic, integer representation and fractional decomposition.