Representation And Coverage Of The Elements In The Residue Classes

The representation and coverage of the elements in the residue classes mod-ulo p is an important problem in number theory where p is a prime.In this paper,we study the representation and coverage of the elements in residue classes by using the estimations of character sums and exponential sums,meanwhile com-bining the method of elementary number theory or some elementary methods in related literatures.The main research contents are as follows:1.We study the problem of the elements in residue classes modulo p repre-sented by the product of two elements in special sets,let p be a prime,Y(?)Zp,we will consider the representation of the residue classes modulo p by xy(mod p)where the value of x is in a special set,y∈Y.A new representation of the residue classes modulo p will also be considered.The cardinalities of some exceptional sets in the residue classes modulo p are given.2.Based on Golomb’s conjectures,we study the problem of the elements in residue classes modulo pα represented by the sum of two elements in special sets,and give some asymptotic formulas for the number of solutions of the congruence.3.We combine the estimates of character sums modulo m with the multi-plication congruence problem to study the representation and coverage of the elements in the residue classes modulo m where m is an arbitrary integer.The asymptotic formula for the cardinality of multiplicative subsets in residue class-es modulo m is proved,thus it is shown that this subset has basic covering properties.This result is a generalization of the previous similar results.