On The Diophantine Problems In Prime Variables

In this dissertation,we mainly consider several Diophantine problems in prime variables,such as the exceptional set for sums of two squares of primes,two cubes of primes and two biquadrates,on the sums of two squares of primes,two cubes of primes and two biquadrates with almost equal variables,Waring-Goldbach problem for unlike powers in prime variables and almost-prime variables,values of binary linear forms at prime arguments,Diophantine in-equalities and equations with Piatetski-Shapiro type variables.There exist total seven chapters in this dissertation.We list the contents of all the chapters as follows:In chapter one,we introduce the history and the development of the problems.Also,current results are listed.Meanwhile,we give our improvement of the results.In chapter two,we use Hardy-Littlewood method to derive the exceptional set for sums of two squares of primes,two cubes of primes and two biquadrates.In chapter three,we consider the same problem,which is in chapter two,with almost equal variabes.In chapter four,we consider two Waring-Goldbach problems for unlike powers in prime variables and almost-prime variables.In chapter five,we consider the values of binary linear forms at prime arguments.In chapter six,we consider the Diophantine inequalities in four and five variables with Piatetski-Shapiro type variables.In chapter seven,we consider the Diophantine equation in five Piatetski-Shapiro type variables.