clc, clearvars

syms p10 dig p x

p=sym(1);

for p10=1:108

   p=p*sym(10); 

   for dig=4:9 

      eq=sym(6)*(10*x+dig)==x+p*dig;

      s=solve(eq,x);

      if s==round(s)

         disp([s dig]) 

         disp([s+p*dig])

         disp((s+p*dig)/(10*s+dig))

         disp(p10)

         disp(' ')         

      end

   end

end

solves the equation for switched digits between 4 and 9, for successive powers of 10 representing how distant the switch is, finding the main body needed and finds 

[67796610169491525423728813559322033898305084745762711864, 4]

4067796610169491525423728813559322033898305084745762711864

6

    57

 

[84745762711864406779661016949152542372881355932203389830, 5]

5084745762711864406779661016949152542372881355932203389830

6

    57

 

[101694915254237288135593220338983050847457627118644067796, 6]

6101694915254237288135593220338983050847457627118644067796

6

    57

 

[118644067796610169491525423728813559322033898305084745762, 7]

7118644067796610169491525423728813559322033898305084745762

6

    57

 

[135593220338983050847457627118644067796610169491525423728, 8]

8135593220338983050847457627118644067796610169491525423728

6

    57

 

[152542372881355932203389830508474576271186440677966101694, 9]

9152542372881355932203389830508474576271186440677966101694

6

    57

    

The first two "solutions" are not really solutions as they introduce a leading zero for the smaller number. The third solution is the lowest actual solution.   

Then subsequent solutions:

N = 1186440677966101694915254237288135593220338983050847457627

becomes

7118644067796610169491525423728813559322033898305084745762

and

N = 1355932203389830508474576271186440677966101694915254237288

becomes

8135593220338983050847457627118644067796610169491525423728

and

N = 1525423728813559322033898305084745762711864406779661016949

becomes

9152542372881355932203389830508474576271186440677966101694

These are 58-digit numbers, the 57 in the program output being the power of 10 for the leftmost position.