1.Introduction



I show the parametric solutions of x^4 + y^4 + z^4 = w^n.

Case of n=2: Diophantus showed a parametric solution.
Case of n=3: I showed a parametric solution.
             x^4 + y^4 + z^4 = nw^3
Case of n=4: Noam Elkies showed there were infinitely many solutions.

In the case of n=5 and 7, parametric solutions and  numerical examples are shown.




2. Result

 


   
    Case 1:  x^4 + y^4 + z^4 = w^5
    
            x= 5580a^8b^2+35760a^7b^3-31248a^5b^5+36540a^6b^4-20880a^3b^7-62580a^4b^6
              +5580a^2b^8-1740a^9b+2980b^9a-298a^10+174b^10
  
            y= -13410a^8b^2-20880a^7b^3+75096a^5b^5+26040a^6b^4-14880a^3b^7+36540a^4b^6
               -13410a^2b^8-1240a^9b-1740b^9a+174a^10+124b^10

            z= -7830a^8b^2+14880a^7b^3+43848a^5b^5+62580a^6b^4-35760a^3b^7-26040a^4b^6
               -7830a^2b^8-2980a^9b+1240b^9a-124a^10+298b^10

            w= 98*(a^2+ab+b^2)^4

   
    Case 2:  x^4 + y^4 + z^4 = w^7
    
            x = -272832a^13b-272832ab^13+13888b^14-33376a^14-100228128a^8b^6
                -27803776a^9b^5+12148864a^11b^3+1263808a^12b^2+41705664a^6b^8
                -66882816a^7b^7+19507488a^10b^4+66818752a^5b^9-5055232a^3b^11
                +19507488a^4b^10-3037216a^2b^12

            y = -194432a^13b-194432ab^13-33376b^14+19488a^14+58522464a^8b^6
                +66818752a^9b^5-7093632a^11b^3-3037216a^12b^2-100228128a^6b^8
                -47663616a^7b^7+13901888a^10b^4-39014976a^5b^9+12148864a^3b^11
                +13901888a^4b^10+1773408a^2b^12
            
            z = -467264a^13b-467264ab^13-19488b^14-13888a^14-41705664a^8b^6
                +39014976a^9b^5+5055232a^11b^3-1773408a^12b^2-58522464a^6b^8
                -114546432a^7b^7+33409376a^10b^4+27803776a^5b^9+7093632a^3b^11
                +33409376a^4b^10-1263808a^2b^12

            w = 392*(a^2+ab+b^2)^4

3. Example


        Case 1: n=5

        (a,b)
       ( 1, 0)          298^4 +          174^4 +          124^4 =              98^5
       ( 2, 1)      5008486^4 +      2084068^4 +      2924418^4 =          235298^5
       ( 3, 1)     76443468^4 +    108116282^4 +     31672814^4 =         2798978^5
       ( 3, 2)    490258926^4 +    236444998^4 +    726703924^4 =        12771458^5
       ( 4, 1)    283390974^4 +   1171803348^4 +    888412374^4 =        19059138^5
       ( 4, 3)   8006223182^4 +  12585945132^4 +  20592168314^4 =       183667778^5
       ( 5, 1)    903436676^4 +   6930003826^4 +   7833440502^4 =        90505058^5
       ( 5, 2)  24578157726^4 +  21994687674^4 +   2583470052^4 =       226717218^5
       ( 5, 3)  71457684476^4 +   3445434726^4 +  74903119202^4 =       564950498^5
       ( 5, 4)  53601123574^4 + 187213005374^4 + 240814128948^4 =      1356892418^5



       


        Case 2: n=7

        (a,b)
      
        ( 1, 0)              33376^4 +             19488^4 +             13888^4 =             392^7
        ( 2, 1)         3862651968^4 +       21749064736^4 +       25611716704^4 =          941192^7
        ( 3, 1)      2077977823264^4 +      752759535808^4 +     1325218287456^4 =        11195912^7
        ( 3, 2)     23700424280288^4 +    27741284238752^4 +     4040859958464^4 =        51085832^7
        ( 4, 1)     50636462845248^4 +    53824111643232^4 +     3187648797984^4 =        76236552^7
        ( 4, 3)   3118894577009344^4 +  2111095853186208^4 +  1007798723823136^4 =       734671112^7
        ( 5, 1)    457688985095328^4 +   922550118182336^4 +   464861133087008^4 =       362020232^7
        ( 5, 2)   3530411806051872^4 +   790887932616096^4 +  4321299738667968^4 =       906868872^7
        ( 5, 3)  11192650517264608^4 + 22741405359898176^4 + 11548754842633568^4 =      2259801992^7
        ( 5, 4) 105230958200303712^4 + 47711778121609504^4 + 57519180078694208^4 =      5427569672^7