1.Introduction

Last time,I gave a parameter solution for 6.4.4.

(-3k2b22-20a12-11a1kb2)6+(4k2b22-12a12+a1kb2)6+(2k2b22-16a12-11a1kb2)6+(-k2b22-8a12-15a1kb2)6
=(4k2b22+8a12+3a1kb2)6+(3k2b22-16a12-a1kb2)6+(k2b22-20a12-13a1kb2)6+(-2k2b22-12a12-17a1kb2)6

This identity is obtained by solving below equation.

(a1a-7c2)6+(a1a-b2b+c2)6+(3a1a+c2)6+(3a1a+b2b+c2)6=(a1a+7c2)6+(a1a-b2b-c2)6+(3a1a-c2)6+(3a1a+b2b-c2)6

I show that there are many parameter solutions besides above one.




2. The generalization of the Chernick's solution (Part 2)


             There is a parameter solution of A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.



             A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.................(1)


Proof.            

             A1=a1a-7c2
             A2=a2a-b1b+c2
             A3=3a1a+c2
             A4=(4a1-a2)a+b1b+c2
             B1=a1a+7c2
             B2=a2a-b1b-c2
             B3=3a1a-c2
             B4=(4a1-a2)a+b1b-c2


      A16+ A26+ A36+ A46 -( B16+ B26+ B36+ B46)

     =240a1ac2(-28c22+a22a2-2baa2b1+b12b2-4a1a2a2+4a1ab1b+7a12a2)(30c22+a22a2-4a1a2a2+9a12a2-2baa2b1+4a1ab1b+b12b2)
   

    (7a12+a22-4a1a2)a2+(4a1b1-2a2b1)ba-28c22+b12b2=0.....................(2)

    We can find infinitely many rational solutions of (2),
    then we obtain infinitely parameter solutions of (1).

    Take b1=1,c2=1
    Then (2) becomes to (7a12+a22-4a1a2)a2+(4a1-2a2)ba-28+b2=0...........(3)

    (a,b)=(3/a1,(3a2-7a1)/a1) is a solution of (3).

    We obtain parameter solution of (2) by using (a,b)=(3/a1,(3a2-7a1)/a1).


             A1=-4k2+(8a2-14a1)k+14a1a2-42a12-4a22

             A2=6k2+(-12a2+46a1)k-46a1a2+6a22+56a12

             A3=10k2+(-20a2+46a1)k-46a1a2+28a12+10a22

             A4=8k2+(-16a2+18a1)k-18a1a2-14a12+8a22

             B1=10k2+(-20a2+42a1)k-42a1a2+56a12+10a22

             B2=4k2+(-8a2+38a1)k-38a1a2+4a22+42a12

             B3=8k2+(-16a2+38a1)k-38a1a2+14a12+8a22

             B4=6k2+(-12a2+10a1)k-10a1a2-28a12+6a22

   
3. Example

       Case: (a1,a2)=(1,1)   
  
            The equation 4a2+2ba-28+b2 has infinitely rational solutions,so 

            the equation (a-7)6+(a-b+1)6+(3a+1)6+(3a+b+1)6=(a+7)6+(a-b-1)6+(3a-1)6+(3a+b-1)6
            has infinitely rational solutions.

            The parameter solution is below.
            (-32-6k-4k2)6+(16+34k+6k2)6+(-8+26k+10k2)6+(-24+2k+8k2)6
           =(24+22k+10k2)6+(8+30k+4k2)6+(-16+22k+8k2)6+(-32-2k+6k2)6

           k

           0    46 +   26 +   16 +   36 =    36 +   16 +   26 +   46
           1    36 +   46 +   26 +   16 =    46 +   36 +   16 +   26
           2    56 +   96 +   76 +   16 =    96 +   76 +   56 +   16
           3   436 +  866 +  806 +  276 =   906 +  676 +  616 +   86
           4   156 +  316 +  326 +  146 =   346 +  246 +  256 +   76
           5   276 +  566 +  626 +  316 =   646 +  436 +  496 +  186
           6   536 + 1096 + 1276 +  696 =  1296 +  836 + 1016 +  436
           7  1356 + 2746 + 3326 + 1916 =  3346 + 2076 + 2656 + 1246
           8    26 +   46 +   56 +   36 =    56 +   36 +   46 +   26
           9  2056 + 4046 + 5186 + 3216 =  5166 + 3016 + 4156 + 2186
          10  1236 + 2396 + 3136 + 1996 =  3116 + 1776 + 2516 + 1376


       Case: (a1,a2)=(1,2)   

            (a-7)6+(2a-b+1)6+(3a+1)6+(2a+b+1)6=(a+7)6+(2a-b-1)6+(3a-1)6+(2a+b-1)6
            where 3a2+b2=28.

            (-30+2k-4k2)6+(-12+22k+6k2)6+(-24+6k+10k2)6+(-18-14k+8k2)6
           =(12+2k+10k2)6+(-18+22k+4k2)6+(-30+6k+8k2)6+(-24-14k+6k2)6


       Case: (a1,a2)=(1,3)   

            (a-7)6+(3a-b+1)6+(3a+1)6+(a+b+1)6-(a+7)6-(3a-b-1)6-(3a-1)6-(a+b-1)6
            where 4a2-2ba+b2=28.

            (-36+10k-4k2)6+(-28+10k+6k2)6+(-20-14k+10k2)6+(4-30k+8k2)6
           =(20-18k+10k2)6+(-36+14k+4k2)6+(-28-10k+8k2)6+(-4-26k+6k2)6

           

       Case: (a1,a2)=(2,1)  

            The equation 21a2+6ba-28+b2 has infinitely rational solutions,so 

            the equation (2a-7)6+(a-b+1)6+(6a+1)6+(7a+b+1)6=(2a+7)6+(a-b-1)6+(6a-1)6+(7a+b-1)6
            has infinitely rational solutions.
           
            
            (-144-20k-4k2)6+(138+80k+6k2)6+(30+72k+10k2)6+(-84+20k+8k2)6
           =(150+64k+10k2)6+(96+68k+4k2)6+(-12+60k+8k2)6+(-126+8k+6k2)6


           k

           0   246 +  236 +   56 +  146 =   256 +  166 +   26 +  216  
           1    36 +   46 +   26 +   16 =    46 +   36 +   16 +   26
           2  1006 + 1616 + 1076 +   66 =  1596 + 1246 +  706 +  436
           3    56 +   96 +   76 +   16 =    96 +   76 +   56 +   16
           4  1446 + 2776 + 2396 +  626 =  2836 + 2166 + 1786 +   16
           5   436 +  866 +  806 +  276 =   906 +  676 +  616 +   86
           6   686 + 1396 + 1376 +  546 =  1496 + 1086 + 1066 +  236
           7   156 +  316 +  326 +  146 =   346 +  246 +  256 +   76
           8   406 +  836 +  896 +  426 =   936 +  646 +  706 +  236
           9   276 +  566 +  626 +  316 =   646 +  436 +  496 +  186
          10  3726 + 7696 + 8756 + 4586 =  8956 + 5886 + 6946 + 2776



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