Small Prime Solution Of General Diagonal Diophantine Equation

The problem of small prime solutions to the Diophantine equation is a key research project in the field of the Goldbach conjecture.As an important extension of Goldbach's problem,it is favored by many analytic number theorists.In this article,we study the prime solution of the integer equations under certain exponents.The prime solution is that every sufficiently large positive integer n that satisfies certain congruence conditions can be represented in the sum of s k-th power of primes.This topic is to study the allowable value of the upper bound of the prime solution to the integer equations,i.e.Pj<<|n|1/k+max |aj|C+?,where 1?j?s,s depends on k.For the case k?3,we get C=1+6·2k+20/3·22k-1+3·2k-1-20.Obviously,C depends on k,i.e.C=C(k)and C(3)=39/22?1.7727….Aiming at the problem of the small prime solution of general diagonal diophantine equation,this article first takes the new idea of Zhao[44]and promotes some important results on the basis of it.At the same time,we add the analytical idea of the classic circle method into it.We get the upper bound of the integral estimate of the exponential sum correspondingly according to Holder's inequality.Finally,we improves and generalizes the result C(3)=2 of Zhao[44]in a way,and also greatly improves the result C(k)=3·2k-1(k?4)due to Yang and Hu[42].In addition,we study the problem of Diophantine approximation by equation of mixed power prime variables,i.e.the sum of one prime,one square,one cube,one fourth and one k-th power of a prime with real coefficients is dense on the entire real number axis.The result of this article improves the result of Mu[35].And we can improve the target value for other equation of mixed power prime variables.