| The functional equations associated with Euler function,Smarandache function S(n),Smarandache LCM function SL(n),pseudo-Smarandache function z(n),the other F.Smarandache multiplicative function S(n),simple number rootfunction sim(n)and p sub-power primitive function equation Sp(n)are the hot discuss topics that a number of known mathematicians have studied in recent years.In this thesis,the several kinds of number theoretic function equations with solvability are studied using elementary and analytic techniques and methods.The main resuits are listed in the following:1.Using the elementary method,we study the integer solutions of the Euler function equation φ(abc)=Aφ(a)+Bφ(b)+Cφ(c)+D(where A,B,C ∈N+,A2+B2=C2,D=0,-(AB+C-1))and φ(φ(n-φ(φ(n))))=k.The thesis obtains that the first type of Euler function equation has 40 and 17 groups of positive integer solutions respectively when(A,B,C)=(3,4,5),D=0,-16.Moreover,the second type of Euler function equation has 33 and 2 positive integer solutions respectively when k=8,10.2.Master the definition of correlation function,the solvability of compound number theory function equation φ(φ(n-S(SL(n))))=M1,M2(here n,M1,M2 ∈ N+)with Smarandache LCM function is studied.It is proved that the equationφ(φ(n-S(SL(n))))=M1 has 8,13,23 and 2 positive integer solutions respectively when M1=2,4,8,10.The equation φ(φ(n-S(SL(n))))=M2 has only n-6 of positive integer solutions when the minimum positive integer is taken of M2(that M2=1),while when the minimum two complete numbers are taken of M2(that M2=6,28),the equation has n=38,57 and n=118,177 of positive integer solutions respectively.3.Using the theory of congruence and the special simplification of the simple root function,the solvability of pseudo Smarandache function and simple number root function of compound number theory function equation Z(nt)=sim(cp(nt))(where n,t ∈N+),Z(nx)=sim(φ(nk))(where n,x,k ∈ N+),and Z(n)=sim(φ2(n))))+1(where n∈N+,the same below)are studied.It is proved that the first type of equations has positive integer solutions n=1,3,5,7,10 and n=1 respectively when t=1,2.Then the second type of equations has only positive integer solutions n=1 when k=2,Y.Moreover,the third type of equation has only positive integer solutions as n=3,15,28.4.Through the above research,combined with Excel’s numerical operations and filtering function,the solvability of the compound number theory function equation Z(n)=Sp(sim(n))(where p=2,3,5)with the simple number root function and the power prime function,and the compound number theory function equation Z(S(n))=S(sim(n))(where p are prime number and n∈N+)is studied.It is proved that the first type of equations has no solution when p=2,while the first type of equations has positive integer solutions n=24,60 and n=11,120 respectively when p-3,5,and the second type of equation has a finite number of positive integer solutions. |
