1.Introduction



I show that there are many parameter solutions of 6.6.6.




2. Theorem


             There are many parameter solutions of A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.



             A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.................(1)
           

Proof.           


             A1=a1a+b1b-c1
             A2=a3a+c3
             A3=a4a+c4
             A4=a1a-b2b+c2
             A5=a2a-b1b+c2
             A6=a2a+b2b+c2
             B1=a1a+b1b+c1
             B2=a3a-c3
             B3=a4a-c4
             B4=a1a-b2b-c2
             B5=a2a-b1b-c2
             B6=a2a+b2b-c2

      A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.............................(2)

      a2=ma1,a3=na1,a4=pa1,c1=qc2,c3=rc2,c4=tc2
      As for this equation (2), the factorization is done as follows.
 
      |(m,n,p,q,r,t)|<6

      (m,n,p,q,r,t)

      (3 1 -1 -2 -6  3) 240a1ac2(-19c22+b22b2+2a1b2ba+4a2a12)(21c22+b22b2+2a1b2ba+6a2a12)
      (3 1 -1 -1 -5  3) 240a1ac2(14c22+6a12a2+2b2ba1a+b22b2)(-12c22+4a12a2+2b2ba1a+b22b2)
      (3 1 -1 2  -2  3) 240a1ac2(5c22+b22b2+2a1ab2b+6a12a2)(-3c22+b22b2+2a1ab2b+4a12a2)
      (3 1 -1 3  -3  1) 240a1ac2(6c22+b22b2+2a1ab2b+6a2a12)(-4c22+b22b2+2a1ab2b+4a2a12)
      (3 2 -4 3   2  2) -240a1ac2(5c22+15a2a12+2ba1ab2+b22b2)(3c22+5a2a12-2ba1ab2-b22b2)
      (3 3 -5 3   2  2) -240a1ac2(3c22-b22b2-2a1ab2b+12a12a2)(5c22+b22b2+2a1ab2b+22a12a2)
      (3 4 -6 3   2  2) -240a1ac2(3c22+21a12a2-2b2a1ba-b22b2)(5c22+31a12a2+2b2a1ba+b22b2)
     



    Take c2=1
    
    Case 1. 4a2a12+2a1b2ba+b22b2=19.....................................(3)
    

    We can find infinitely many rational solutions of (3),then we obtain infinitely parameter solutions of (2).

    Take x=2a1a,y=b2b 

    Then (3) becomes to x2+xy+y2=19........................................(4)

    (x,y)=(3,2) is a solution of (4).

    We obtain parameter solution of (3) by using (a,b)=(3/(2a1),2/b2).
         
                 
             A1=7k2b22-4a12
             A2=-9k2b22-32kb2a1-68a12
             A3=3k2b22+20kb2a1+44a12
             A4=15k2b22+20kb2a1-28a12
             A5=11k2b22-20kb2a1-52a12
             A6= k2b22-44kb2a1-36a12
             B1=-k2b22-16kb2a1-36a12
             B2=15k2b22+16kb2a1+28a12
             B3=-9k2b22-4kb2a1-4a12
             B4=11k2b22+12kb2a1-44a12
             B5=7k2b22-28kb2a1-68a12
             B6=-3k2b22-52kb2a1-52a12
   
   Case 2. 4a2a12+2a1b2ba+b22b2=12

                   
             A1=4k2b22-4kb2a1-8a12
             A2=-8k2b22-28kb2a1-56a12
             A3=4k2b22+20kb2a1+40a12
             A4=12k2b22+12kb2a1-24a12
             A5=8k2b22-20kb2a1-40a12
             A6=-12a1(3kb2+2a1)
             B1=-12(kb2+2a1)a1
             B2=12k2b22+12kb2a1+24a12
             B3=-8k2b22-4kb2a1-8a12
             B4=8k2b22+4kb2a1-40a12
             B5=4k2b22-28kb2a1-56a12
             B6=-4k2b22-44kb2a1-40a12
 


   
   Case 3. 5a2a12-2ba1ab2-b22b2=-3


             A1=-4k2b22-20kb2a1+32a12
             A2=8k2b22-8kb2a1-16a12
             A3=-4k2b22+40kb2a1-28a12
             A4=12k2b22-24kb2a1+12a12
             A5=8k2b22-20kb2a1-4a12
             A6= -24a12
             B1=8k2b22+4kb2a1-28a12
             B2=-24(kb2-a1)a1
             B3=-12k2b22+24kb2a1+12a12
             B4=8k2b22-32kb2a1+32a12
             B5=4k2b22-28kb2a1+16a12
             B6=-4k2b22-8kb2a1-4a12

   
   Case 4. 12a2a12-2ba1ab2-b22b2=-3
                   

             A1=-5k2b22-20kb2a1+76a12
             A2=7k2b22-16kb2a1-36a12
             A3=-k2b22+48kb2a1-68a12
             A4=9k2b22-28kb2a1+28a12
             A5= 5k2b22-20kb2a1-12a12
             A6=-k2b22+4kb2a1-60a12
             B1=7k2b22+4kb2a1-68a12
             B2=-k2b22-32kb2a1+60a12
             B3=-9k2b22+32kb2a1+28a12
             B4=5k2b22-36kb2a1+76a12
             B5= k2b22-28kb2a1+36a12
             B6=-5k2b22-4kb2a1-12a12
   
   Case 5. 21a2a12-2ba1ab2-b22b2=-3
                    

             A1=8k2b22+30kb2a1-198a12
             A2=-10k2b22+36kb2a1+90a12
             A3=12(-7kb2+15a1)a1
             A4=-12k2b22+48kb2a1-72a12
             A5=-6k2b22+30kb2a1+36a12
             A6=2k2b22-12kb2a1+162a12
             B1=-10k2b22-6kb2a1+180a12
             B2=2k2b22+60kb2a1-162a12
             B3=12k2b22-60kb2a1-72a12 
             B4=-6k2b22+60kb2a1-198a12
             B5=-6(-7kb2+15a1)a1
             B6= 8k2b22+36a12



   
3. Example


        Case 1: (a1,b2)=(1,1)

        k

        1     36+ 1096+  676+   76+  616+  796 =   536+  596+  176+  216+  896+ 1076
        2     16+   76+   46+   36+   26+   56 =    36+   56+   26+   16+   46+   76
        3    596+ 2456+ 1316+ 1676+  136+ 1596 =   936+ 2116+  976+  916+  896+ 2356
        4    276+  856+  436+  736+  116+  496 =   296+  836+  416+  456+  176+  776
        5    576+ 1516+  736+ 1496+  416+  776 =   476+ 1616+  836+  976+  116+ 1296
        6    316+  736+  346+  796+  286+  336 =   216+  836+  446+  536+   26+  596
        7   3396+ 7336+ 3316+ 8476+ 3476+ 2956 =  1976+ 8756+ 4736+ 5796+  796+ 5636
        8    376+  756+  336+  916+  416+  276 =   196+  936+  516+  636+  136+  556
        9   5636+10856+ 4676+13676+ 6596+ 3516 =  2616+13876+ 7696+ 9556+ 2476+ 7636
       10    876+ 1616+  686+ 2096+ 1066+  476 =   376+ 2116+ 1186+ 1476+  446+ 1096


       Case 2: (a1,b2)=(1,1)

       k

       1     26+  236+  166+   06+  136+  156 =    96+  126+   56+   76+  206+  226
       2     06+   36+   26+   16+   16+   26 =    16+   26+   16+   06+   26+   36
       3     46+  536+  346+  306+   76+  336 =   156+  426+  236+  116+  266+  526
       4     56+  376+  236+  276+   16+  216 =    96+  336+  196+  136+  136+  356
       5     66+  336+  206+  286+   56+  176 =    76+  326+  196+  156+   86+  306
       6     76+  326+  196+  306+   86+  156 =    66+  336+  206+  176+   56+  286
       7    406+ 1616+  946+ 1626+  536+  696 =   276+ 1746+ 1076+  956+  146+ 1366
       8     96+  336+  196+  356+  136+  136 =    56+  376+  236+  216+   16+  276
       9    706+ 2396+ 1366+ 2646+ 1076+  876 =   336+ 2766+ 1736+ 1616+   46+ 1906
      10    226+  716+  406+  816+  356+  246 =    96+  846+  536+  506+   46+  556


      Case 3: (a1,b2)=(1,1)
  
      k

      1     16+   26+   16+   06+   26+   36 =    26+   06+   36+   16+   16+   26 
      2     26+   06+   36+   16+   16+   26 =    16+   26+   16+   06+   26+   36
      3     86+   46+   76+   66+   16+   36 =    76+   66+   36+   16+   46+   86
      4    286+  206+  176+  276+  116+   66 =   296+  186+  216+   86+   86+  256
      5     76+   66+   36+   86+   46+   16 =    86+   46+   76+   36+   16+   66
      6    586+  566+  176+  756+  416+   66 =   716+  306+  696+  326+   26+  496
      7    386+  406+   76+  546+  316+   36 =   496+  186+  516+  256+   26+  326
      8    326+  366+   36+  496+  296+   26 =   436+  146+  476+  246+   46+  276
      9    596+  706+   16+  966+  586+   36 =   826+  246+  936+  496+  116+  506
     10   1426+ 1766+   76+ 2436+ 1496+   66 =  2036+  546+ 2376+ 1286+  346+ 1216


     Case 4: (a1,b2)=(1,1)
  
     k

     1    176+  156+   76+   36+   96+  196 =   196+   96+  176+  156+   36+   76
     2     26+   56+   36+   16+   46+   76 =    46+   16+   76+   36+   26+   56
     3    296+  216+  676+  256+  276+  576 =    76+  456+  436+  136+  396+  696
     4     76+   16+   96+   56+   16+   56 =    56+   76+   16+   16+   56+   96
     5   1496+  596+ 1476+ 1136+  136+  656 =  1276+ 1256+  376+  216+  796+ 1576
     6    286+  156+  236+  236+   66+   96 =   266+  216+  136+   56+  126+  276
     7   1036+  656+  736+  916+  316+  276 =  1016+  716+  636+  236+  376+  956
     8   1016+  716+  636+  956+  376+  236 =  1036+  656+  736+  276+  316+  916
     9   5096+ 3876+ 2836+ 5056+ 2136+ 1056 =  5356+ 3096+ 4136+ 1576+ 1356+ 4536
    10    266+  216+  136+  276+  126+   56 =   286+  156+  236+   96+   66+  236


    Case 5: (a1,b2)=(1,1)
  
    k

    1    406+  296+  246+   96+  156+  386 =   416+  256+  306+  366+  126+  116 
    2    536+  616+   66+  126+  366+  736 =   646+  176+  726+  516+   36+  346
    3     16+   36+   26+   16+   26+   46 =    26+   16+   46+   26+   16+   36
    4    256+  376+  786+  366+  306+  736 =    26+  556+  606+  276+  396+  826
    5    386+   56+  606+  336+   96+  386 =   256+  476+  186+  126+  306+  596
    6     56+   16+   66+   46+   06+   36 =    46+   56+   06+   16+   36+   66
    7   1016+  376+ 1026+  816+  126+  446 =   886+  896+  246+  186+  516+ 1076
    8   2776+ 1316+ 2466+ 2286+  546+  976 =  2546+ 2236+ 1086+  516+ 1236+ 2746
    9    206+  116+  166+  176+   56+   66 =   196+  156+  106+   46+   86+  196
   10    416+  256+  306+  366+  126+  116 =   406+  296+  246+   96+  156+  386


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