1.Introduction

Chernick(1937) gave a parameter solution for 6.4.4.

(-10-9k-5k2)6+(-4+7k+9k2)6+(-8-5k+7k2)6+(-6-13k+k2)6=(4+5k+9k2)6+(-6+5k+7k2)6+(-10-7k+5k2)6+(-8-15k-k2)6([1].Wolfram)

This identity is obtained by solving below equation.

(a-7)6+(a-2b+1)6+(3a+1)6+(3a+2b+1)6=(a+7)6+(a-2b-1)6+(3a-1)6+(3a+2b-1)6([2].Piezas)

I show that there are many parameter solutions besides the Chernick's one.

[1].Wolfram: Multigrade Eauation
[2].Titus Piezas: Binary Quadratic Forms as Equal Sums of Like Powers

2. The generalization of the Chernick's solution


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

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


             A1=-3k2b22-20a12-11a1kb2
             A2=4k2b22-12a12+a1kb2
             A3=2k2b22-16a12-11a1kb2
             A4=-k2b22-8a12-15a1kb2
             B1=4k2b22+8a12+3a1kb2
             B2=3k2b22-16a12-a1kb2
             B3=k2b22-20a12-13a1kb2
             B4=-2k2b22-12a12-17a1kb2




Proof.            

             A1=a1a-7c2
             A2=a1a-b2b+c2
             A3=3a1a+c2
             A4=3a1a+b2b+c2
             B1=a1a+7c2
             B2=a1a-b2b-c2
             B3=3a1a-c2
             B4=3a1a+b2b-c2


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

     =-240a1ac2(30c22+b22b2+2a1ab2b+6a12a2)(28c22-b22b2-2a1ab2b-4a12a2)

    Take c2=1

    28-b22b2-2a1ab2b-4a12a2=0.............................(2)

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

    Take x=2a1a,y=b2b 

    Then (2) becomes to x2+xy+y2=28......................(3)

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

    We obtain parameter solution of (2) by using (a,b)=(1/a1,4/b2).


             A1=-3k2b22-20a12-11a1kb2
             A2=4k2b22-12a12+a1kb2
             A3=2k2b22-16a12-11a1kb2
             A4=-k2b22-8a12-15a1kb2
             B1=4k2b22+8a12+3a1kb2
             B2=3k2b22-16a12-a1kb2
             B3=k2b22-20a12-13a1kb2
             B4=-2k2b22-12a12-17a1kb2





   
3. Example

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

      (-3k2-20-11k)6+  (4k2-12+k)6+  (2k2-16-11k)6+  (-k2-8-15k)6 = 
      (4k2+8+3k)6   +  (3k2-16-k)6+  (k2-20-13k)6  +  (-2k2-12-17k)6

            k

            1   346+  76+ 256+ 246 =  156+ 146+ 326+ 316 
            2    96+  16+  56+  76 =   56+  16+  76+  96
            3   806+ 276+ 316+ 626 =  536+  86+ 506+ 816
            4    46+  26+  16+  36 =   36+  16+  26+  46
            5   506+ 316+  76+ 366 =  416+ 186+ 206+ 496
            6   976+ 696+  56+ 676 =  856+ 436+ 316+ 936
            7  2446+1916+  56+1626 = 2256+1246+ 626+2296
            8   256+ 216+  26+ 166 =  246+ 146+  56+ 236
            9  3626+3216+ 476+2246 = 3596+2186+ 566+3276
           10  2156+1996+ 376+1296 = 2196+1376+ 256+1916


       Case: (a1,b2)=(2,1)  
  
      (-3k2-80-22k)6+  (4k2-48+2k)6+  (2k2-64-22k)6+  (-k2-32-30k)6 =
      (4k2+32+6k)6  +  (3k2-64-2k)6+  (k2-80-26k)6 +  (-2k2-48-34k)6

            k
 
            1    56+  26+  46+  36 =   26+  36+  56+  46 
            2   346+  76+ 256+ 246 =  156+ 146+ 326+ 316
            3  1736+  66+1126+1316 =  866+ 436+1496+1686
            4    96+  16+  56+  76 =   56+  16+  76+  96
            5  2656+ 626+1246+2076 = 1626+  16+1856+2686
            6   806+ 276+ 316+ 626 =  536+  86+ 506+ 816
            7  1276+ 546+ 406+ 976 =  906+ 236+ 716+1286
            8    46+  26+  16+  36 =   36+  16+  26+  46
            9  5216+2946+1006+3836 = 4106+1616+2336+5166
           10   506+ 316+  76+ 366 =  416+ 186+ 206+ 496