Small Solutions Of Cubic Equations With Prime Variables In Arithmetic Progressions

The principal purpose of this paper is to consider the bounds of solutions of the cubic equationwith the prime variables in arithmetic progressions modulo k > 1. By the use of the circle method, and then treating the singular integral carefully, the best qualitative results are obtained. A natural necessary and sufficient condition for the solvability to the equation is given explicitly.The paper consists of seven sections. In the first section, we introduce the background of this problem, the development of Baker problem and the latest results in this respect. At the same time, we also introduce our main results. In section 2, we define some necessary notations and give the outline of the proof of the main results. The weighted number of solutions of equation, r(b), is expressed as the sum of a integral on [0,1] and a remainder term. Using the circle method, the integral on the interval [0, 1] is divided into the integrals on major arcs and minor arcs respectively, J and J . The section 3 gives a transformation for Sj(a) andsimplifies /. The major term of / is divided into M1,M2 and M$. In section 4,by the use of the method of the singular integral and the precise computations ofsingular series, we estimate MI . The computations of singular series are tedious but coincident and insure the conditions listed in section one to be sufficient and necessary. In the fifth section, we deal with general singular series, by which we get an estimate of 5^r In section 6, we complete the estimation for the major arcintegrals /. In the last section, we prove minor arcs integral / has a smallerupper bound. So we complete the whole proof. In addition, considering the fact the theory of cubic residue is not often used, for the brief of the paper and the convenience of understanding, we provide in the appendix some knowledge of cubic residue which may be used in the paper .