1.Introduction

      (2018.11.17)
      Added the solutions of Taxicab(6,5,2),Taxicab(6,5,3), and Taxicab(6,5,4).
      
      Generalized Taxicab Number Taxicab(k,j,n) is the smallest number which can be expressed as
      the sum of j kth powers in n different ways.([1].Cluade)

      When k=5 and 6 and j=3,only 2-way sum is known.
      Taxicab(5,3,2) =                    1375298099 = 67^5 + 28^5 + 24^5 = 62^5 + 54^5 + 3^5.
      Taxicab(6,3,2) =                     160426514 = 23^6 + 15^6 + 10^6 = 22^6 + 19^6 + 3^6.

      When k=7 and 8 and j=4,only 2-way sum is known. 
      Taxicab(7,4,2) =              2056364173794800 = 149^7 + 123^7 + 14^7 + 10^7 = 146^7 + 129^7 + 90^7 + 15^7.
      Taxicab(8,4,2) =  9299777658352887974911487459 = 3113^8 + 2012^8 + 1953^8 + 861^8 = 2823^8 + 2767^8 + 2557^8 + 1128^8.

      When k=9 and 10 and j=5,only 2-way sum is known.
      Taxicab(9,5,2)  =         356099046047451719360 = 192^9 + 101^9 + 91^9 + 30^9 + 26^9 = 180^9 + 175^9 + 116^9 + 17^9 + 12^9.
      Taxicab(10,5,2) =          63130456038464578275 = 95^10 + 71^10 + 32^10 + 28^10 + 25^10 + 16^10 = 92^10 + 85^10 + 34^10 + 34^10 + 23^10 + 5^10.

      The case of Taxicab(k,3,n) and n > 2 is not known yet where k = 5 and 6.
      So,I searched the case of Taxicab(k,4,n) where k = 5 and 6 and the case of Taxicab(k,5,n) where k = 7 and 8.
      I searched the case of Taxicab(7,5,3) and Taxicab(8,5,3),but I could not find a solution.
      Is there a solution of Taxicab(7,5,3) and Taxicab(8,5,3)?

      [1].Calude,C. S.; Calude, E.; and Dinneen, M. J. "What Is the Value of Taxicab(6)?" J. Uni. Comp. Sci. 9, 1196-1203, 2003. 
      [2].http://mathworld.wolfram.com/DiophantineEquation5thPowers.html
          http://mathworld.wolfram.com/DiophantineEquation6thPowers.html
          http://mathworld.wolfram.com/DiophantineEquation7thPowers.html


2. Result

 
       Taxicab(5,4,2)=          4062500  =  20^5 +   15^5 +   10^5 +    5^5
                                         =  19^5 +   16^5 +   14^5 +    1^5


       Taxicab(5,4,3)=       1479604544  =  66^5 +   44^5 +   36^5 +   18^5
                                         =  64^5 +   51^5 +   36^5 +   13^5
                                         =  61^5 +   52^5 +   48^5 +    3^5

            
                                                 
       Taxicab(6,4,2)=       200408169   =  24^6 +   14^6 +   11^6 +    4^6
                                         =  22^6 +   21^6 +   10^6 +    8^6


       Taxicab(6,4,3)=   1885800643779   = 111^6 +   49^6 +   34^6 +    1^6
                                         = 110^6 +   69^6 +   43^6 +    7^6
                                         = 109^6 +   77^6 +   25^6 +   18^6

       Taxicab(6,5,2)=        17446027   =  14^6 +   13^6 +   13^6 +    8^6 +   3^6
                                         =  16^6 +    9^6 +    7^6 +    5^6 +   4^6
                                         
       Taxicab(6,5,3)=     25581719370   =  50^6 +   46^6 +   28^6 +    9^6 +   3^6
                                         =  51^6 +   44^6 +   30^6 +    7^6 +   2^6
                                         =  52^6 +   42^6 +   26^6 +   15^6 +  11^6
                                         
                                         
       Taxicab(6,5,4)=  68168072613226   = 183^6 +  173^6 +  124^6 +   74^6 +  34^6
                                         = 191^6 +  156^6 +  130^6 +   85^6 +  22^6
                                         = 191^6 +  162^6 +  100^6 +   85^6 +  74^6
                                         = 200^6 +  123^6 +   92^6 +   68^6 +  17^6
                                         
       Taxicab(7,5,2)=     10500437728   =  26^7 +   21^7 +   18^7 +   12^7 +  11^7
                                         =  24^7 +   23^7 +   22^7 +   10^7 +   9^7


       Taxicab(8,5,2)=  11714114636483   =  43^8 +   20^8 +   11^8 +   10^8 +   1^8
                                         =  41^8 +   35^8 +   32^8 +   28^8 +   5^8