Research On The Solutions Including Diophantine Equation And Euler Function And Related Problems

In the process of learning number theory function,the Diophantine equation and Euler function have been paid much attention.Previously,many domestic and foreign scholars have studied these equations and functions in depth and promoted the development of number theory.In this paper,we use the methods of elementary number theory,algebraic number theory,analytical number theory,and other methods to discuss Diophantine equation,Euler functions,and related equation problems.The main contents are as followsIn the first part,The solvability of an Diophantine equation x2+1024=4y9 is analyzed and all positive integer solutions are givenIn the second part,we discuss the problem of positive integer solution of two kinds of equations involving Euler function,One of them studies the solvability of the equations ?(mn)-A?(m)+B?(n)+C,and gives all positive integer solutions of these The other is the solvability of the equation ?(abc)-A?(a)+B?(b)+C?(c)-D,all positive integer solutions of the equation are givenIn the third part,The solvability of two equations involving Euler function and Smarandache function S(SL(n13,14))=?2(n)is analyzed,and all positive integer solutions are given.