1.Introduction



I show the parametric solutions of x^4 + y^4 + z^4 = nw^3 about several n.


In the case of n=1,2, and 3, parametric solutions and  numerical examples are shown.


Smallest solutions of x^4 + y^4 + z^4 = nw^3 in the case of n < 10 are as follows.

       n       x      y     z      w

       1      76     72     4    392
       2      19     18     1     49
       3      228   216    12   1176
       4      38     36     2     98
       5      380   360    20   1960
       6      57     54     3    147
       7      32     20    12     56
       8      76     72     4    196 
       9      106    91    80    297




2. Theorem

      There are parametric solutions of x^4 + y^4 + z^4 = nw^3,n=1,2, and 3.

Proof.           
 
     I prove the case of n=2,since this is the simplest case.

     First,we use the identity x^4 + y^4 + (x+y)^4 = 2(x^2+xy+y^2)^2.

     To find (x,y) of (x^2+xy+y^2)=A^3,we use the u such that u^3=1.

     Let (x-yu)(x-yu^2)={(a-bu)(a-bu^2)}^3.

     So,(x-yu)=(a-bu)^3=(-3a^2b-3ab^2)u+a^3-3ab^2-b^3.

     Then,we obtain x = a^3-3ab^2-b^3,y = 3a^2b+3ab^2,z = x+y = a^3-b^3+3a^2b.

     Similarly,we can obtain the parametric solutions of the case n=1 and 3.



   
    Case 1:  x^4 + y^4 + z^4 = w^3
    

  
          
             x = -76a^6+1080a^4b^2+1520a^3b^3+60a^2b^4-432ab^5-76b^6-24a^5b
              
             y = 4a^6-1140a^4b^2-80a^3b^3+1080a^2b^4+456ab^5+4b^6-432a^5b

             z = -72a^6-60a^4b^2+1440a^3b^3+1140a^2b^4+24ab^5-72b^6-456a^5b

             w = 392(b^2+ba+a^2)^4

   
    Case 2:  x^4 + y^4 + z^4 = 2w^3
    
  

          
             x = a^3-3ab^2-b^3
              
             y = 3a^2b+3ab^2

             z = a^3-b^3+3a^2b

             w = (a^2+ab+b^2)^2
  
    Case 3:  x^4 + y^4 + z^4 = 3w^3
    
   

          
             x = -228a^6+3240a^4b^2+4560a^3b^3+180a^2b^4-1296ab^5-228b^6-72a^5b
              
             y = 12a^6-3420a^4b^2-240a^3b^3+3240a^2b^4+1368ab^5+12b^6-1296a^5b

             z = -216a^6-180a^4b^2+4320a^3b^3+3420a^2b^4+72ab^5-216b^6-1368a^5b

             w = 1176(a^2+ab+b^2)^4

   
3. Example


        Case 1: n=1

        (a,b)
       ( 1, 0)       76^4+       4^4+      72^4 =             392^3 
       ( 2, 1)    23108^4+   27212^4+    4104^4 =          941192^3
       ( 3, 1)    66452^4+  185468^4+  119016^4 =        11195912^3
       ( 3, 2)   573476^4+  394124^4+  179352^4 =        51085832^3
       ( 4, 1)    37044^4+  703836^4+  666792^4 =        76236552^3
       ( 4, 3)  4332308^4+ 2229692^4+ 2102616^4 =       734671112^3
       ( 5, 1)   398236^4+ 1980716^4+ 2378952^4 =       362020232^3
       ( 5, 2)  2832516^4+ 5062284^4+ 2229768^4 =       906868872^3
       ( 5, 3)  9333716^4+ 7926044^4+ 1407672^4 =      2259801992^3
       ( 5, 4) 19333364^4+ 8114396^4+11218968^4 =      5427569672^3



       


        Case 2: n=2

        (a,b)
       ( 1, 0)        1^4+       0^4+       1^4 = 2*              1^3 
       ( 2, 1)        1^4+      18^4+      19^4 = 2*             49^3
       ( 3, 1)       17^4+      36^4+      53^4 = 2*            169^3
       ( 3, 2)       17^4+      90^4+      73^4 = 2*            361^3
       ( 4, 1)       51^4+      60^4+     111^4 = 2*            441^3
       ( 4, 3)       71^4+     252^4+     181^4 = 2*           1369^3
       ( 5, 1)      109^4+      90^4+     199^4 = 2*            961^3
       ( 5, 2)       57^4+     210^4+     267^4 = 2*           1521^3
       ( 5, 3)       37^4+     360^4+     323^4 = 2*           2401^3
       ( 5, 4)      179^4+     540^4+     361^4 = 2*           3721^3

       

        Case 3: n=3

        (a,b)

       ( 1, 0)      228^4+      12^4+     216^4 = 3*           1176^3 
       ( 2, 1)    69324^4+   81636^4+   12312^4 = 3*        2823576^3
       ( 3, 1)   199356^4+  556404^4+  357048^4 = 3*       33587736^3
       ( 3, 2)  1720428^4+ 1182372^4+  538056^4 = 3*      153257496^3
       ( 4, 1)   111132^4+ 2111508^4+ 2000376^4 = 3*      228709656^3
       ( 4, 3) 12996924^4+ 6689076^4+ 6307848^4 = 3*     2204013336^3
       ( 5, 1)  1194708^4+ 5942148^4+ 7136856^4 = 3*     1086060696^3
       ( 5, 2)  8497548^4+15186852^4+ 6689304^4 = 3*     2720606616^3
       ( 5, 3) 28001148^4+23778132^4+ 4223016^4 = 3*     6779405976^3
       ( 5, 4) 58000092^4+24343188^4+33656904^4 = 3*    16282709016^3