1.Introduction



I show the parametric solutions of x4+y4+z4=nw2 about several n.


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

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

S.Realis
(5a^4+4a^3+9a^2+10a+5)^4 + (5a^4+10a^3+9a^2+4a+5)^4 + (5a^4+16a^3+27a^2+16a+5)^4 = 3((a^2+a+1)(25a^6+75a^5+222a^4+319a^3+222a^2+75a+25))^2

Smallest solutions of x4+y4+z4=nw2 in the case of n < 20 are as follows.

       n       x       y       z      w

       1      20      15      12     481
       2       2       1       1       3
       3       1       1       1       1
       4      40      30      24     962
       8       4       2       2       6
       9      60      45      36    1443
      11       5       5       3      11
      12       2       2       2       2
      16      40      30      24     481
      17      22      17       6     137
      18       2       1       1       1
      19      15      15       1      73


Above equation doesn't have a solution in the case of n=5,6,7,10,13,14,15 mod 16.

2. Theorem

             x,y,z,w,n: integer
             Assume one solution a14+a24+a34=na42 is known.

             There are parametric solutions of x4+y4+z4=nw2.


             x4+y4+z4=nw2.................(1)
           

Proof.           
             a1,a2,a3,a4,a,b: integer
             

             Set

             x=a1t+a
             y=a2t+b
             z=a3t
             w=a4t2+gt+h......................(2)

             Substitute (2) to (1),then (1) becomes to (3).

             (4a23b+4a13a-2na4g)t3+(-ng2+6a22b2+6a12a2-2na4h)t2+(-2ngh+4a2b3+4a1a3)t+a4+b4-nh2..........(3)
             Equating to zero the coefficient of t3,we get g.

             g= 2*(a23b+a13a)/(na4)
             
             Similarly,we get h and t.

             h = (-2a26b2-4a23ba13a-2a16a2+3a22b2na42+3a12a2na42)/(n2a43)

             t = -1/4*(a4n3a46+b4n3a46-18a22b2n2a44a12a2+24a25b3a13ana42+24a23ba15a3na42+12a16a2a22b2na42
                 -9a24b4n2a44-9a14a4n2a44+12a26b2a12a2na42-16a23ba19a3+12a18a4na42-16a29b3a13a-24a26b2a16a2+12a28b4na42-4a212b4-4a112a4)
               /(na42*(2a29b3+6a26b2a13a+6a23ba16a2-3a25b3na42-3a23ba12a2na42+2a19a3-3a13aa22b2na42-3a15a3na42+a2b3n2a44+a1a3n2a44))

             Substitute g,h and t to (2).

             x = 4a113a4-4a19a4na42+16a23ba110a3+24a26b2a17a2+16a29b3a14a+6a22b2n2a44a13a2-12a17a2a22b2na42
               + 12a26b2a13a2na42-24a25b3a14ana42-3a15a4n2a44+3a1a4n3a46-a1b4n3a46+4a1a212b4-12a1a28b4na42
               + 9a1a24b4n2a44+8ana42a29b3-12an2a44a25b3-12a3n2a44a23ba12+4an3a46a2b3 

             y = 4a213b4+6a23b2n2a44a12a2-24a24ba15a3na42+12a16a2a23b2na42-12a27b2a12a2na42-12a2a18a4na42
               + 9a2a14a4n2a44+8bna42a19a3-12b3n2a44a13aa22-12bn2a44a15a3+4bn3a46a1a3+16a24ba19a3-4a29b4na42
               + 24a27b2a16a2+16a210b3a13a-3a25b4n2a44-a2a4n3a46+3a2b4n3a46+4a2a112a4 

             z = -a3*(a4n3a46+b4n3a46-18a22b2n2a44a12a2+24a23ba15a3na42+12a16a2a22b2na42-4a212b4
               - 4a112a4+12a26b2a12a2na42+24a25b3a13ana42+12a18a4na42-16a23ba19a3+12a28b4na42-24a26b2a16a2
               - 16a29b3a13a-9a14a4n2a44-9a24b4n2a44)

             w = a4*(288a111a5n3a46a2b3-280a17a5n4a48a2b3-96a2^15b5a1a3n2a44-8a13ab7n5a410a2+88a23b5n5a410a1a3
               + 288a211b5n3a46a1a3-8a23ba7n5a410a1-96a1^15a5a2b3n2a44+88a13a5n5a410a2b3-280a27b5n4a48a1a3
               - 24b4n3a46a112a4+ 48b4n4a48a18a4-336b5n4a48a23a15a3+16b6n4a48a16a2a22+12b6n5a410a22a12a2
               + 264b8n3a46a212-26a4n5a410a24b4-119b8n4a48a28+192a5n3a46a29b3a13+22a8n5a410a14+b8n6a412+48a4n4a48a28b4
               + 720a6n3a46a26b2a16+768a7n3a46a23ba19+16a6n4a48a26b2a12-24a4n3a46a212b4-336a5n4a48a25b3a13
               - 60a6n4a48a16a22b2+264a8n3a46a112+12a6n5a410a22b2a12-184a7n4a48a23ba15-119a8n4a48a18+2a4n6a412b4
               - 1344a23ba113a7n2a44-1824a29b3a17a5n2a44+1152a23ba1^17a7na42+1152a2^15b5a15a3na42-2256a26b2a110a6n2a44
               - 1344a2^18b6a16a2-384a221b7a13a-1728a211b5n2a44a15a3-2592a28b4n2a44a18a4-1728a25b3n2a44a111a5
               + 288a210b6n3a46a12a2+864a27b5n3a46a15a3+288a22b2n3a46a110a6+864a28b4n3a46a14a4+192a120a8na42
               - 432a22b2n2a44a1^14a6-384a121a7a23b-432a2^14b6n2a44a12a2+864a24b4n3a46a18a4-74a24b4n4a48a14a4
               + 864a25b3n3a46a17a5+768b7n3a46a29a13a+22b8n5a410a24+192b5n3a46a23a19a3+720b6n3a46a26a16a2
               - 184b7n4a48a25a13a-26b4n5a410a14a4-60b6n4a48a26a12a2+3840a19a3a211b5na42+2880a112a4a28b4na42
               - 2256a16a2a210b6n2a44+1152a1^15a5a25b3na42-3360a212b4a112a4-1824a19a3a27b5n2a44-2688a1^15a5a29b3+192a1^18a6a22b2na42
               + 192a220b8na42+2880a16a2a2^14b6na42-312a1^16a8n2a44-696a112a4a24b4n2a44-48a224b8+2880a212b4a18a4na42+3840a29b3a111a5na42
               + 192a2^18b6a12a2na42-48a124a8+2880a26b2a1^14a6na42+1152a2^17b7a13ana42-2688a2^15b5a19a3-1344a1^18a6a26b2-696a212b4a14a4n2a44-312a2^16b8n2a44
               - 1344a213b7a13an2a44+a8n6a412)
    
    
    
    Case 1. n=2
    

    Set (a1,a2,a3,a4)=(2,1,1,3),then we obtain a parametric solution of (1).

          
             x = 12928b^3a-7296b^2a^2+832ba^3-208a^4-6256b^4    
              
             y = 16456b^4+1912a^4+22272b^2a^2-9664ba^3-30976b^3a

             z = 1912a^4-3128b^4-3328b^3a+10176b^2a^2-5632ba^3

             w = -728838144ab^7+3655872a^8+193597632b^8-1208683008b^5a^3+810681984b^4a^4-29246976ba^7+132742656a^6b^2+1213088256b^6a^2-386998272a^5b^3

   
    Case 2. n=3
    

    Set (a1,a2,a3,a4)=(1,1,1,1),then we obtain a parametric solution of (1).

          
             x = 46a^4-92ba^3+78b^2a^2-32b^3a+22b^4   
              
             y = 46b^4 +78b^2a^2-32ba^3+22a^4-92b^3a

             z = 22a^4 +22b^4 +114b^2a^2-56ba^3-56b^3a

             w = -12768a^5b^3-12768b^5a^3-5136ba^7+14028b^4a^4 +10248b^6a^2+10248a^6b^2+1284a^8+1284b^8-5136ab^7


  
    Case 3. n=11
    

    Set (a1,a2,a3,a4)=(5,5,3,11),then we obtain a parametric solution of (1).

          
             x = 13245205990a^4-46902287500ba^3-41337083680b^3a+62513643750b^2a^2+11722914670b^4
              
             y = -41337083680ba^3+62513643750b^2a^2+11722914670a^4-46902287500b^3a+13245205990b^4

             z = 7033748802a^4+7033748802b^4-25715625000b^3a-25715625000ba^3+39933933750b^2a^2

             w = 3557132992348890891332a^4b^4+68828742151520431916a^8-487455680959776430000ab^7
                -2908613564855678020000b^5a^3-2908613564855678020000a^5b^3-487455680959776430000ba^7
                +1549677364327195895000b^6a^2+1549677364327195895000a^6b^2+68828742151520431916b^8

   
3. Example


        Case 1: n=2

        (a,b)

       ( 1,-3)     1444^4+    3751^4+      83^4 = 2*       10057653^2
       ( 1,-2)      362^4+     959^4+      47^4 = 2*         656883^2
       ( 1,-1)       43^4+     127^4+      28^4 = 2*          11493^2
       ( 1, 0)       26^4+     239^4+     239^4 = 2*          57123^2
       ( 2,-3)    37270^4+  101749^4+   11461^4 = 2*     7386758811^2
       ( 2,-2)       43^4+     127^4+      28^4 = 2*          11493^2
       ( 2,-1)       22^4+      85^4+      37^4 = 2*           5211^2
       ( 2, 0)       26^4+     239^4+     239^4 = 2*          57123^2
       ( 2, 1)       46^4+      23^4+     121^4 = 2*          10467^2
       ( 3,-3)       43^4+     127^4+      28^4 = 2*          11493^2
       ( 3,-2)    18370^4+   62131^4+   21379^4 = 2*     2759021451^2
       ( 3,-1)     1172^4+    5669^4+    3167^4 = 2*       23825493^2
       ( 3, 0)       26^4+     239^4+     239^4 = 2*          57123^2
       ( 3, 1)       43^4+      28^4+     127^4 = 2*          11493^2
       ( 3, 2)      610^4+    1133^4+    2179^4 = 2*        3487851^2

        Common factors were removed. 


        Case 2: n=3

        (a,b)

       ( 1,-3)    367^4 +   703^4 +   451^4 = 3*     318201^2
       ( 1,-2)    115^4 +   187^4 +   139^4 = 3*      24297^2
       ( 1,-1)      1^4 +     1^4 +     1^4 = 3*          1^2
       ( 1, 0)     23^4 +    11^4 +    11^4 = 3*        321^2
       ( 1, 1)     11^4 +    11^4 +    23^4 = 3*        321^2
       ( 2,-3)   4631^4 +  6311^4 +  5303^4 = 3*   30752241^2
       ( 2,-2)      1^4 +     1^4 +     1^4 = 3*          1^2
       ( 2,-1)    187^4 +   115^4 +   139^4 = 3*      24297^2
       ( 2, 0)     23^4 +    11^4 +    11^4 = 3*        321^2
       ( 2, 1)      1^4 +     1^4 +     1^4 = 3*          1^2
       ( 2, 2)     11^4 +    11^4 +    23^4 = 3*        321^2
       ( 3,-3)      1^4 +     1^4 +     1^4 = 3*          1^2
       ( 3,-2)   6311^4 +  4631^4 +  5303^4 = 3*   30752241^2
       ( 3,-1)    703^4 +   367^4 +   451^4 = 3*     318201^2
       ( 3, 0)     23^4 +    11^4 +    11^4 = 3*        321^2
       ( 3, 1)    187^4 +   139^4 +   115^4 = 3*      24297^2
       ( 3, 2)    115^4 +   139^4 +   187^4 = 3*      24297^2
       ( 3, 3)     11^4 +    11^4 +    23^4 = 3*        321^2




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