On A Binary Additive Equation

A binary additive equation with prime numbers is studied in the first part of this paper.The problem of a diophantine equation is one of the most important classical problems. Its study has pretty good significance in theory.Let 1 < c < 17/16, and N is a sufficiently large integer. Lporta[21] proved that, almost all n∈(N, 2N] can be represented as n∈[p₁^c] + [p₂^c], where P₁, p₂≤N^1/c are prime numbers and [x] denotes the integer part of x. His method also yields an asymptotic formula for the number of representations of these n.In this paper we improve the range of c in Lporta[21]. Let 1 < c < 8/7, and N is a sufficiently large integer. We prove that, almost all n∈(N, 2N] can be representde as n= [p₁^c] + [P₂^c], where P₁, P₂≤N^1/c are prime numbers and [x] denotes the integer part of x.Letwhere n∈(N, 2N], N is a sufficiently large integer, p₁, p₂≤N^1/c are prime numbers and [x] denotes the integer part of x. Lporta[21] made full use of the estimate of an exponential sum in Lporta and Tolev[10]where c is a fixed real number satisfying 1 < c < 17/16, P=N^1/cΩ₂=(ω, 1-ω),ω=P^1-c-η,η<0.001. In this paper we imitate the method of the estimate of an exponential sum in Wenguang Zhai[20], then we obtainwhere c is a fixed real number satisfying 17/16≤c<8/7, P=N^1/c,Ω₂=(ω,1-ω),ω=p^1-c-ηη<(8/7-c)×10^-3. E=exp(-A(log N)^1/3-ε,εis an arbitrary small positive number.In this paper we have two main results:(1)Let 10 and 0<ε<1/3 be arbitrary constants. Then(2)Let10 and 0<ε<1/3 be arbitrary constants. Then for all n∈E (N, 2N] but O(Nexp(-B(logN)^1/3-ε) exceptions, the equation n= [p₁^c]+[p₂^c] is solvable with primes P1, P2≤N^1/c and we haveA binary additive equation with square-free numbers is studied in the second part.Let 1 < c < 4/3, and N is a sufficiently large integer. Deshouillers[22] proved that, N can be represented as N=[n^c] + [m^c], where n, m are integers. Subsequently, the range for c in this result was extended by Gritsenko[23]å’ŒKonyagin[24]. In particular, the latter author showed that N = [n^c] + [m^c] has solutions in integers n, m for 1 < c < 3/2 and N sufficiently large.In this paper we study the solution problem of N = [n^c]+[m^c] for n, m are square-free numbers.We obtain the main result:Let then we have asymptotic formula where 1 < c < 6/5,â–³=1/c-5/6.