The Solvability For The Number Theory Equation S(SL(n~k))=?_e(n)(k=1,2)

Since the generalized Euler function was proposed in 2007,it is an open problem to determine its explicit computing formula.Until now,the explicit computing formula of ?_e(n)is given only when e=3,4,6.Based on the explicit computing formula of?_e(n)(e=1,2),and the properties for S(n),SL(n)and ?_e(n),the solvability of the number theory equation S(SL(n~k))=?_e(n)(k=1,2)is studied,and then all the positive integer solutions are determined.By using elementary methods and techniques,this thesis discusses the solvability of the number theory equation S(SL(n~k))=?_e(n)(k=1,2),and the all positive integer solutions are determined when e=3,4,6.