On The Representation Of Numbers As Sums Of Squares And Cubes

In the 1770s.the British mat hematician E.Waring proposed a conjecture tjhat for every positive integer k that is not less than 3,there exists interger s,such that every positive interger can be represented as the sum of s k-th powers,and we use g(k)to denote the smallest such s.Hilbert proved that for every natural number k,g(k)exists.The modern problem is that for each k,we want to find the smallest natural number G(k)such that every sufficiently large natural number is the sum of at most G(k)k-th powers.The function G(k)has only been determined for two values of k,namely G(2)＝4,by Lagrange in 1770,and G(4)=16,by Davenport.The first explicit general upper bound for G(k),namely G(k)≤(k-2)2k-1+5 was obtained by Hardy and Littlewood.The record on the upper bound of G(k)is due to Wooley,who proved in 1995 that G(k)≤k(log k+log log k 2+O(loglog k/log Ak))In addition,let rk,s(n)denote the number of representations of n as the sum of s k-th powers,one of the most important problem is to find the asymptotic formula for rk,s(n).The study of related problems has developed rapidly after the introduction of the circle method by Hardy and Littlewood.The circle method can not only cleal wit h the homogeneous Waring’s problem,but also can be used to deal with the mixed power Waring’s problem.Vaughan established asymptotic formulas for a wide class of Waring’s type.problem.Let R(n)denote the number of represen-tations of positive number nas the sum of two squares and four cubes.As a corollary to Vaughan’s asymptotic formula,there exists N0 such that R(n)>0 whenever n≥N0.We provide an explicit value of N0 in this thesis.We used the circle method,exponential sum estimation and methods from elenentary number theory and analysis in this thesis.There are four chapters in this thesis.In Chapter one.we.briefly introduce some notations and Vaughan’s research results,as well as the main results of this thesis.In Chapter two,we mainly provide some estimations of the commonly used function involved in this paper.In Chapter three,we provide integral estimations over the major arcs and the minor arcs.We give a specific lower bound of the singular series and the value of No.In Chapter four,we introduced the outline of verifying R(n),where n≤108.