Solution Of Two Kinds Of Diophantine Equations And The Study On A Type Of Smarandache Function Equation

Diophantine equation and the equation problem involving Smarandache function are two hot research topics in the number theory,their research achievements have greatly enriched the theoretical contents of the number theory and played a vital role in other subjects as well.Therefore,it is necessary to continue to study the two kinds of the equations.On this thesis,the problem of the integer solution of two kinds of Diophantine equations and a type of Smarandache function equation are studied by using the methods of elementary and algebraic number theory.The main research results are as follows:Firstly,the problem of integer solution of Diophantine equation x²+D=4y⁵ is discussed in the integer ring of quadratic Euclid???.Above all,we prove the equation in the integer ring of virtual quadratic Euclid ??? has only integer solution?x,y?=?±1,1?when D=3.Next,we prove the equation in the corresponding integer ring of real quadratic Euclid???has no integer solution when D=-3,-6,-7,-37.Secondly,we discuss the solvability problem of Diophantine equation x²+4^k=y⁹ in the Gauss integer ring,and prove that the equation has only integer solution?x,y?=?±16,2?when k=4,however,it has no integer solution when k=3,5.Thirdly,combining with C++ and python programs,we discuss the solvability problem of a kind of equation ?₂?n?=S?n^k?,k=,7,8,9 involving Smarandache function ?₂?n? and generalized Euler function ?₂?n?,at last,we get all the positive integer solutions of the equations.