1.Introduction

      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)

      The case of Taxicab(4,2,n) and n > 2 is not known yet.
      So,I searched the case of Taxicab(4,3,n).
   
      I produced (x^4+y^4+z^4,x,y,z) and found the duplicated elements. 

      Taxicab(4,3,10:12) means that Taxicab(4,3,10) = Taxicab(4,3,11) = Taxicab(4,3,12).

      [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. Result

 
       Taxicab(4,3,2)=           2673  =   7^4 +    4^4 +    2^4
                                       =   6^4 +    6^4 +    3^4

              
                                                 
       Taxicab(4,3,3)=         811538  =  29^4 +   17^4 +   12^4
                                       =  28^4 +   21^4 +    7^4
                                       =  27^4 +   23^4 +    4^4
                                            

          
       Taxicab(4,3,4)=      5978882  =  48^4 +   25^4 +   23^4
                                     =  47^4 +   32^4 +   15^4
                                     =  45^4 +   37^4 +    8^4
                                     =  43^4 +   40^4 +    3^4
                                                                                      

       Taxicab(4,3,5)=    137149922  = 105^4 +   56^4 +   49^4 
                                     = 104^4 +   65^4 +   39^4 
                                     =  99^4 +   80^4 +   19^4 
                                     =  96^4 +   85^4 +   11^4 
                                     =  91^4 +   91^4 +    0^4 




       Taxicab(4,3,6)=    292965218  = 127^4 +   66^4 +   61^4
                                     = 126^4 +   77^4 +   49^4
                                     = 123^4 +   89^4 +   34^4
                                     = 121^4 +   94^4 +   27^4
                                     = 119^4 +   98^4 +   21^4
                                     = 111^4 +  109^4 +    2^4

                              
       Taxicab(4,3,7)=    779888018  = 167^4 +   38^4 +    9^4 
                                     = 162^4 +   89^4 +   73^4 
                                     = 161^4 +   98^4 +   63^4 
                                     = 158^4 +  111^4 +   47^4 
                                     = 157^4 +  114^4 +   43^4 
                                     = 147^4 +  133^4 +   14^4 
                                     = 142^4 +  139^4 +    3^4 

       Taxicab(4,3,8)=   5745705602  = 267^4 +  145^4 +  122^4 
                                     = 265^4 +  163^4 +  102^4 
                                     = 263^4 +  173^4 +   90^4 
                                     = 262^4 +  177^4 +   85^4 
                                     = 255^4 +  197^4 +   58^4 
                                     = 250^4 +  207^4 +   43^4 
                                     = 243^4 +  218^4 +   25^4 
                                     = 233^4 +  230^4 +    3^4 

       Taxicab(4,3,9)= 49511121842   = 458^4 +  233^4 +  225^4 
                                     = 457^4 +  255^4 +  202^4 
                                     = 455^4 +  273^4 +  182^4 
                                     = 447^4 +  310^4 +  137^4 
                                     = 442^4 +  325^4 +  117^4 
                                     = 441^4 +  322^4 +  175^4 
                                     = 427^4 +  357^4 +   70^4 
                                     = 413^4 +  378^4 +   35^4 
                                     = 403^4 +  390^4 +   13^4 
                                    

       Taxicab(4,3,10:12)=281539574498 = 707^4 +  371^4 +  336^4 
                                       = 704^4 +  411^4 +  293^4 
                                       = 701^4 +  432^4 +  269^4 
                                       = 699^4 +  443^4 +  256^4 
                                       = 696^4 +  457^4 +  239^4 
                                       = 693^4 +  469^4 +  224^4 
                                       = 679^4 +  511^4 +  168^4 
                                       = 664^4 +  543^4 +  121^4 
                                       = 656^4 +  557^4 +   99^4 
                                       = 649^4 +  568^4 +   81^4 
                                       = 632^4 +  591^4 +   41^4 
                                       = 616^4 +  609^4 +    7^4 

       

       Taxicab(4,3,13:16)=7865870969138 = 1626^4 +  829^4 +  797^4 
                                        = 1622^4 +  911^4 +  711^4 
                                        = 1621^4 +  922^4 +  699^4 
                                        = 1619^4 +  941^4 +  678^4 
                                        = 1611^4 +  997^4 +  614^4 
                                        = 1601^4 + 1047^4 +  554^4 
                                        = 1579^4 + 1126^4 +  453^4 
                                        = 1578^4 + 1129^4 +  449^4 
                                        = 1571^4 + 1149^4 +  422^4 
                                        = 1553^4 + 1194^4 +  359^4 
                                        = 1549^4 + 1203^4 +  346^4 
                                        = 1503^4 + 1289^4 +  214^4 
                                        = 1494^4 + 1303^4 +  191^4 


     Taxicab(4,3,17)= 47580188090162    = 2548^4 + 1365^4 + 1183^4 
                                        = 2545^4 + 1413^4 + 1132^4 
                                        = 2540^4 + 1467^4 + 1073^4 
                                        = 2535^4 + 1508^4 + 1027^4 
                                        = 2525^4 + 1572^4 +  953^4 
                                        = 2523^4 + 1583^4 +  940^4 
                                        = 2515^4 + 1623^4 +  892^4 
                                        = 2487^4 + 1732^4 +  755^4 
                                        = 2457^4 + 1820^4 +  637^4 
                                        = 2443^4 + 1855^4 +  588^4 
                                        = 2405^4 + 1937^4 +  468^4 
                                        = 2372^4 + 1997^4 +  375^4 
                                        = 2353^4 + 2028^4 +  325^4 
                                        = 2327^4 + 2067^4 +  260^4 
                                        = 2303^4 + 2100^4 +  203^4 
                                        = 2293^4 + 2113^4 +  180^4 
                                        = 2212^4 + 2205^4 +    7^4 


     Taxicab(4,3,18)= 67780725801698    = 2786^4 + 1407^4 + 1379^4 
                                        = 2782^4 + 1521^4 + 1261^4 
                                        = 2779^4 + 1561^4 + 1218^4 
                                        = 2769^4 + 1651^4 + 1118^4 
                                        = 2757^4 + 1726^4 + 1031^4 
                                        = 2751^4 + 1757^4 +  994^4 
                                        = 2743^4 + 1794^4 +  949^4 
                                        = 2718^4 + 1889^4 +  829^4 
                                        = 2709^4 + 1918^4 +  791^4 
                                        = 2654^4 + 2061^4 +  593^4 
                                        = 2639^4 + 2093^4 +  546^4 
                                        = 2626^4 + 2119^4 +  507^4 
                                        = 2593^4 + 2179^4 +  414^4 
                                        = 2581^4 + 2199^4 +  382^4 
                                        = 2531^4 + 2274^4 +  257^4 
                                        = 2507^4 + 2306^4 +  201^4 
                                        = 2457^4 + 2366^4 +   91^4 
                                        = 2434^4 + 2391^4 +   43^4 


     Taxicab(4,3,19)= 101635786393778   = 3173^4 +  722^4 +  171^4 
                                        = 3083^4 + 1542^4 + 1541^4 
                                        = 3082^4 + 1609^4 + 1473^4 
                                        = 3078^4 + 1691^4 + 1387^4 
                                        = 3066^4 + 1813^4 + 1253^4 
                                        = 3059^4 + 1862^4 + 1197^4 
                                        = 3037^4 + 1978^4 + 1059^4 
                                        = 3017^4 + 2058^4 +  959^4 
                                        = 3002^4 + 2109^4 +  893^4 
                                        = 2983^4 + 2166^4 +  817^4 
                                        = 2967^4 + 2209^4 +  758^4 
                                        = 2957^4 + 2234^4 +  723^4 
                                        = 2934^4 + 2287^4 +  647^4 
                                        = 2891^4 + 2373^4 +  518^4 
                                        = 2863^4 + 2422^4 +  441^4 
                                        = 2793^4 + 2527^4 +  266^4 
                                        = 2763^4 + 2566^4 +  197^4 
                                        = 2727^4 + 2609^4 +  118^4 
                                        = 2698^4 + 2641^4 +   57^4 

3. Consideration

It is interesting Taxicab(4,3,10) = Taxicab(4,3,11) = Taxicab(4,3,12) and
Taxicab(4,3,13) = Taxicab(4,3,14) = Taxicab(4,3,15) = Taxicab(4,3,16).

As candidate for Taxicab(4,3,10) which isn't contained in Taxicab(4,3,12) are larger than Taxicab(4,3,12).

For example, one of candidate for Taxicab(4,3,10) = 364765611938 is larger than true Taxicab(4,3,10) = 281539574498.

So,Taxicab(4,3,10) ,Taxicab(4,3,11) and Taxicab(4,3,12) have same value.



I found that a^4+b^4+(a+b)^4=2(a^2+ab+b^2) gave smallest solutions for n to 18 from 3.

For example for n=3,we search the integer solutions of a^2+ab+b^2=637=7^2*13.

Like the above example,prime factors must be the form 3k+1.

We find three solutions {a,b}={23,4},{21,7},and {17,12}.

These solutions give the relation for 27^4 + 23^4 + 4^4 = 28^4 + 21^4 + 7^4 = 29^4 + 17^4 + 12^4 = 811538.

About n=19,only 3173^4 +  722^4 +  171^4 doesn't satisfy the identity of a^4+b^4+(a+b)^4=2(a^2+ab+b^2).

Other solutions for n=19 using identity of a^4+b^4+(a+b)^4=2(a^2+ab+b^2) are not the smallest solutions.

I think it is interesting.

Why a^4+b^4+(a+b)^4=2(a^2+ab+b^2) doesn't give a smallest solution for n=19?