Determining the rank of a number in the residue number system

In this article, the formulation and proof of the theorem on the difference in the ranks of the numbers represented in the Residue Number System is carried out. A method is proposed that allows to reduce the amount of necessary calculations and increases the speed of calculating the rank of a number relative to the method for calculating the rank of a number based on the approximate method. To find the rank of a number in the method for calculating the rank of a number based on the approximate method, it is necessary to calculate n operations with numbers exceeding the modulus value; in the proposed method, it is necessary to calculate n·(n−1)/2 operations not exceeding the value of the module.