1.Introduction


This time, I treat the application of a4+b4+(a+b)4=2(a2+ab+b2)2.

We can find the solution for a14+b14+c14 = a24+b24+c24 =..... = an4+bn4+cn4 according to in 
thinking about the rational solution of f(a,b)=a2+ab+b2.

I chose the solutions as little as possible though I could make the solutions
for how much as well.




2. Result

 
             
              
             215532774+  253016734+   37483964   =   225157034+  246431714+   21274684
                                                 =   234767964+  238220434+    3452474
                                                 =   239505074+  233403034+    6102044
                                                 =   245459484+  226415214+   19044274
                                                 =   262387724+  196790794+   65596934
                                                 =   271758714+  159306834+  112451884
                                                   
                                                    

          
             2801926014+ 3289217494+  487291484 =   2927041394+ 3203612234+  276570844
                                                =   3051983484+ 3096865594+   44882114
                                                =   3113565914+ 3034239394+   79326524
                                                =   3190973244+ 2943397734+  247575514
                                                =   3411040364+ 2558280274+  852760094
                                                =   3532863234+ 2070988794+ 1461874444
                                                =   1115147814+ 3476637294+ 2361489484
                                                =   1678011794+ 3548582314+ 1870570524
                                                =   1824509964+ 3549861774+ 1725351814
                                                =   1929930734+ 3545686694+ 1615755964
                                                =   2120376444+ 3526278214+ 1405901774
                                            

         Extra(5th power, using other identity).
                                                                             

         17907961895+29846603155+11938641265+35815923785+ 5969320635+41785244415

        =24730042615+23024522435+ 7674840815+40079724235+ 8527600905+39226964145

        =28009889115+19744675935+ 6428499145+41326065905+10561105735+37193459315

        =31001955535+16752609515+ 5776761905+41977803145+12901434915+34853130135

        =31790102895+15964462155+ 5691677815+42062887235+13604498185+34150066865

        =32360001315+15394563735+ 5655145865+42099419185+14137864655+33616700395

        =33395928935+14358636115+ 5646654655+42107910395+15165301065+32589263985
        
        =34475872215+13278692835+ 5725674895+42028890155+16324264585+31430300465