1.Introduction

2020.9.6 added.

We show numerical solutions of x4+y4+z4=nw2 for n<1000 with (x,y,z)<10000.
This equation doesn't have a solution in the case of n=5,6,7,10,13,14,15 mod 16.
Furthermore, it has no solution in the case that n has a factor 5^(2m+1).(ex: n=35,65,105,115...)
If n is not squarefree, it is excluded.

No solution was found for n=203,377,609,667,754,899. 

  n     x     y     z      w

  1    12    20    15     481
  2     2     1     1       3
  3     1     1     1       1
 11     3     5     5      11
 17    17    22     6     137
 19     1    15    15      73
 33     1     2     2       1
 41   292   820   275  106509
 43     7     9     9      19
 51   563   565   425   67881
 57     1     4     4       3
 59    97    45     5    1253
 66     8     5     5       9
 67     5     7     7       9
 73     5     2     2       3
 82   205    17     8    4641
 83     3     1     1       1
 89    29    60    40     427
 91    51    55    45     469
 97   190    54    33    3679
107    11    89    89    1083
113     3     2     2       1
114     8    25    25      83
123    41   139    19    1749
129     7   520   400   27663
131  2605  2567  1285  838929
137    28     4     3      67
139     1     5     5       3
146    64     5     5     339
161    22    10     5      39
163     1     3     3       1
177    16    35    28     111
178     2     3     3       1
179   147 11375  3925 9739417
187    85   103    19     939
193    19    24     6      49
194    12   335    65    8063
201    40   506    55   18061
209   208  1025    90   72737
211   767  8775  3885 5401969
217    11    12     6      13
219   583   535   295   30597
226     9    20    15      31
227    53    25     3     191
233    17    26    16      51
241    34    45    30     161
249   445   320   268   14843
251   319   215   165    7261
257    23     2     2      33
258     4     1     1       1
259  1955   615   507  239183
267     7     1     1       3
273     4     2     1       1
274     8   245    35    3627
281    76   360   235    8411
283   439  1365  1287  148637
291    97  1205     5   85121
299   155   451   145   11907
307    97     1     1     537
313    52    30    21     163
321    55   130    26     959
322    87    33    22     427
323    65   163    17    1497
329    46    90    65     517
331    59   545   305   17109
337    26    42     9     103
339   407 11375  2725 7039103
347   195   209    19    3109
353    13     2     2       9
354     2    65    25     227
371    37    25    15      79
379   381  1335   775   96893
386   416  1005   505   53749
393     5    16     4      13
401    30   112    35     631
402    38    13    13      73
403   251   211   103    3879
409    61    90    30     443
411   301  2105   235  218629
417    85    58    22     391
418     4     3     3       1
419   575   325   111   16967
427    37   153    45    1139
433    87    56    48     409
434    13    60     5     173
443   445   217   203    9867
449   268  1255   340   74607
451   897  8405  2595 3341791
457     8     2     1       3
466   125  1845   108  157691
467    29     7     5      39
473    53    90    32     397
481   367  9810  1410 4388929
482    33    52    13     133
483    55  1333   527   81833
489   463   400   220   12293
491  2237  2145  1345  317461
497    13    32    12      47
498    17     5     2      13
499    43  1005   705   50393
514   935   514   115   40287
521    86  4255  1510  799461
523   137   281    23    3549
537    53   484   308   10907
546    23    25    20      39
547   603   411    53   17143
553    16     6     3      11
561    34    10     5      49
562     4     7     7       3
563    13    11     7       9
569    50    35    14     117
571  2873  8175  2475 2829661
577    37     4     4      57
579    11   115    65     577
587    53    19    11     117
593     3     4     4       1
601  5078  3160  2735 1168491
611    25   175   133    1431
617    15    32    32      59
619  3855  2343   275  636773
626    71   240   145    2459
627     5     1     1       1
633    53    28    20     117
641  1720  1450   813  145711
642    74    53    53     267
643    61    59    11     201
649   840   834   425   39533
651   445   349   145    9149
658    52    51    27     149
659    91    35    25     327
673    50    78    33     257
674    90   475   133    8723
681    19    20    10      21
683   343   235     3    4973
689    21    20    10      23
691   485   445   427   13599
697    82   204    27    1597
699    43   685   475   19693
706    25   850   493   28689
707     5     3     1       1
713    16    22    11      21
721  3056  2325  1230  405799
723    71   149    25     847
731  7735  5345  1083 2452621
737    51   124    52     583
739  1587   805   765   98057
753    11    34    26      49
761    14    40    15      59
763    43    19    17      69
769    64   270    15    2633
771  1109   505   235   45279
777    41    40    32      91
779   743 10795  2665 4182987
786    94   115    35     569
787     5     3     3       1
793   459   426   326   10571
802    53    54     9     143
803    79   131    79     681
809  1355   338   100   64677
811  2965  2455  1487  382251
817    16     5     2       9
818    94    13     1     309
827    41    15     7      59
834    19    10     5      13
843    83    41    41     251
849    65   700   434   18017
851    75   973   175   32471
857    35    68    66     221
859   571   455   145   13197
866    77   100    45     401
881  3570  2785   604  502801
883    49    13     1      81
889   655   510   492   18683
897     5     4     2       1
898    26    25    19      33
907   275   209   169    3051
913    16   225    96    1703
914   802   645   185   25363
921   335   436   340    8211
923    23    57    19     109
929    34    65    40     153
937    14   111    24     403
939  1223   515   365   49763
946   745   490   137   19671
947    87   193   113    1303
953    53     6     2      91
962     5     4     3       1
969   670   340   301   15173
971    65   127    45     539
977    43    48    44     113
978    59    52    11     141
979  3215  1591   635  340353
987     5   143   127     829
993    11    16     4       9
994    48   135    25     583






We 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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