We Propose a New Number Representation and Arithmetic For the Elementsof the Ring of Integers Modulo P. the So- Called Polynomial Modular Numbersystem (Pmns) Allows For Fast Polynomial Arithmetic and Easy Parallelization.The Most Important Contribution of This Paper Is the FundamenTal Theorem of Amodular Number System, Which Provides a Bound For the Coefficients of the Polynomialsused to RepResent the Set However, We Alsopropose a Complete Set of Algorithms to Perform the Arithmetic Operations Overa Pmns, Which Make This System of Practical Interest For People Concerned Aboutefficient Implementation of ModuLar Arithmetic.