The Quadratic Waring's Problem In Short Intervals

The Waring's problem was put forward by E.Waring in his book "Meditations Al-gebraicae" in 1770.Each natural number can be expressed as the sum of the square of four numbers,the sum of the cube of the nine non-negative integers,the sum of the four times of nineteen integers,and so on.In China,the history of the Waring's problem,to a certain extent,represents the early history of the study of Chinese number theorist,so it is of great significance.In this paper we mainly studied the quadratic Waring's problem in short interval-s.Using the exponential sums and the mean value theorem,we obtained the result that each sufficiently large positive integer n can be written as n=x₁²+x₂²+x₃²+x₄²+x₅²,whereand xj subject to X-Y<xj?X+Y,X=[???],and xj are integers,1 ?j?5.The article mainly focuses on the quadratic Waring's problem in short interval-s,which will be divided into four chapters.In the first chapter,we briefly introduced the history of number theory and the Waring's problem,and the main result of the paper.In the second chapter,we mainly introduced some definitions and basic theorems in this paper.In the third chapter,after a brief introduction to the Hardy-Littlewood circle method,we gave the division of the major and minor arc and dealt with the minor arc.In the fourth chapter,we first dealt with the major arc,,and then we gave the complete proof of the theorem.