1.Introduction

Sinha(1966) gave a solution for 8.7.7 by using Diophantine system.

Sinha's Theorem

        if a1n +a2n +a3n = b1n +b2n +b3n (n=2,4) then
        
        (2a1)n +(2a2)n +(b1 +b2 +b3)n +(2a3)n +(b1 -b2 +b3)n +(-b1 +b2 +b3)n +(b1 +b2 -b3)n
        =
        (a1 -a2 +a3)n +(-a1 +a2 +a3)n +(2b3)n +(a1 +a2 +a3)n +(2b1)n +(2b2)n +(a1 +a2 -a3)n

                                                                     (n=1,2,4,6,8)


 Sinha didn't give the parameter solution for 8.7.7,so I will show it concretely.


2. Theorem


             There is a parameter solution of A1n+ A2n+ A3n+ A4n+ A5n+ A6n+ A7n = B1n+ B2n+ B3n+ B4n+ B5n+ B6n+ B7n.

             A1n+ A2n+ A3n+ A4n+ A5n+ A6n+ A7n = B1n+ B2n+ B3n+ B4n+ B5n+ B6n+ B7n..........(1)
             (n=1,2,4,6,8)

             A1=4a4+2ba3+2ca3-4b2a2-4ba2c-4c2a2-2b3a-4ab2c-4abc2-2c3a+6b3c-6bc3
             A2=-2ba3-4b2a2-4ba2c+2b3a-4ab2c-4abc2+4b4+2b3c-4b2c2-2bc3-6ca3+6c3a
             A3=-4b2a2-4c2a2-3b3c+3b3a-4b2c2-3ba3+3bc3+3ca3-3c3a+2a^4-6ba2c-6ab2c-6abc2+2b4+2c4
             A4=-2ca3-4ba2c-4c2a2-4ab2c-4abc2+2c3a-2b3c-4b2c2+2bc3+4c4-6b3a+6ba3
             A5=-4c2a2-5b3c+b3a-ba3+5*bc3-3ca3+3c3a+2a4-2ba2c-2ab2c-2abc2-2b4+2c4
             A6=3b3c+5b3a-4b2c2-5ba3-3bc3+ca3-c3a-2a4-2ba2c-2ab2c-2abc2+2b4+2c4
             A7=-4b2a2-b3c-3b3a+3ba3+bc3+5ca3-5c3a+2a4-2ba2c-2ab2c-2abc2+2b4-2c4
             B1=-4c2a2+b3c-5b3a+5ba3-bc3+3ca3-3c3a+2a4-2ba2c-2ab2c-2abc2-2b4+2c4
             B2=-3b3c-b3a-4b2c2+ba3+3*bc3-5ca3+5c3a-2a4-2ba2c-2ab2c-2abc2+2b4+2c4
             B3=-2ca3-4ba2c-4c2a2-4ab2c-4abc2+2c3a-2b3c-4b2c2+2bc3+4c4+6b3a-6ba3
             B4=-4b2a2-4c2a2+3b3c-3b3a-4b2c2+3ba3-3bc3-3ca3+3c3a+2a^4-6ba2c-6ab2c-6abc2+2b4+2c4
             B5=4a4+2ba3+2ca3-4b2a2-4ba2c-4c2a2-2b3a-4ab2c-4abc2-2c3a-6b3c+6bc3
             B6=-2ba3-4b2a2-4ba2c+2b3a-4ab2c-4abc2+4b4+2b3c-4b2c2-2bc3+6ca3-6c3a
             B7=-4b2a2+5b3c+3b3a-3ba3-5bc3-ca3+c3a+2a4-2ba2c-2ab2c-2abc2+2b4-2c4




Proof.            
 
 1. Solving for a12 +a22 +a32 = b12 +b22 +b32.

    Set a1=ax+s1,a2=bx+s2,a3=cx+s3,b1=ax-s2,b2=bx-s3,b3=cx-s1.............................(2)

    Take s3=-(as1+bs2+as2+cs1)/(c+b) then

    a12 +a22 +a32 - ( b12 +b22 +b32) = 0

 2. Solving for a14 +a24 +a34 = b14 +b24 +b34.

   f3:coefficient of x3
   f2:coefficient of x2
   f1:coefficient of x

   a14 +a24 +a34 -( b14 +b24 +b34)
   =f3x3+f2x2+f1x

   First,to satisfy f3 = 0,we find (s1,s2) and s3 below.

   
   s1=b2c-b2a-bc2+bac+a3-c2a

   s2=b2c+b2a-bc2-bac-a3+c2a

   s3=(-b3c-b3a+ba3+bc3-ca3+c3a)/(c+b)

   Next,to satisfy f2x2+f1x = 0,we find that x=-f1/f2.

   x=2/3(a3-ba2-a2c-b2a-bac-c2a+b3-b2c-bc2+c3)/(c+b)

   Substitute (s1,s2,s3,x) to (2), then we obtain (a1,a2,a3,b1,b2,b3) below.


    a1=2a4+ba3+ca3-2b2a2-2ba2c-2c2a2-b3a-2ab2c-2abc2-c3a+3b3c-3bc3
    a2=-ba3-2b2a2-2ba2c+b3a-2ab2c-2abc2+2b4+b3c-2b2c2-bc3-3ca3+3c3a
    a3=-ca3-2ba2c-2c2a2-2ab2c-2abc2+c3a-b3c-2b2c2+bc3+2c4-3b3a+3ba3
    b1=2a4+ba3+ca3-2b2a2-2ba2c-2c2a2-b3a-2ab2c-2abc2-c3a-3b3c+3bc3
    b2=-ba3-2b2a2-2ba2c+b3a-2ab2c-2abc2+2b4+b3c-2b2c2-bc3+3ca3-3c3a
    b3=-ca3-2ba2c-2c2a2-2ab2c-2abc2+c3a-b3c-2b2c2+bc3+2c4+3b3a-3ba3.......................(3)

    From Sinha's Theorem

    A1=2a1
    A2=2a2
    A3=b1+b2+b3
    A4=2a3
    A5=b1-b2+b3
    A6=-b1+b2+b3
    A7=b1+b2-b3

    B1=a1-a2+a3
    B2=-a1+a2+a3
    B3=2b3
    B4=a1+a2+a3
    B5=2b1
    B6=2b2
    B7=a1+a2-a3........................................................................(4)

    Substitute (3) to (4), then we obtain a parameter solution of (1).

 

   
3. Example
         (n=1,2,4,6,8)

     (1,-2,-5)   220n+  316n+  525n+  614n+  481n+  115n+   71n=  355n+  259n+  596n+   39n+  410n+   44n+  575n
     (2,-4,-5)     5n+  127n+  175n+  178n+   92n+   10n+   93n=  155n+   23n+   82n+   28n+  185n+   83n+  150n
     (2,-1,-4)    73n+  166n+  210n+  173n+   88n+  103n+   19n=  133n+   40n+  191n+   33n+  107n+  122n+  206n
     (4,-2,-5)   361n+  355n+  429n+  116n+  244n+  160n+   25n=   55n+   61n+  404n+  300n+  269n+  185n+  416n
      (4,1,-2)   191n+  122n+   33n+  107n+  133n+  206n+   40n=   88n+   19n+   73n+  210n+  173n+  166n+  103n
      (4,2,-5)    95n+  107n+   50n+   38n+  123n+   65n+    8n=  120n+   82n+   58n+   13n+  115n+   73n+   25n
       (4,3,2)    11n+  162n+  135n+   79n+   69n+   94n+   28n=   36n+  115n+  163n+  126n+   41n+   66n+   47n
     (5,-2,-4)    58n+   73n+   13n+  115n+  120n+   25n+   82n=  123n+    8n+   95n+   50n+   38n+  107n+   65n
      (5,2,-4)   404n+  185n+  300n+  269n+   55n+  416n+   61n=  244n+   25n+  361n+  429n+  116n+  355n+  160n
      (5,2,-1)   596n+   44n+   39n+  410n+  355n+  575n+  259n=  481n+   71n+  220n+  525n+  614n+  316n+  115n
      (5,4,-2)    82n+   83n+   28n+  185n+  155n+  150n+   23n=   92n+   93n+    5n+  175n+  178n+  127n+   10n
       (5,4,2)     8n+  335n+  275n+  157n+  150n+  187n+   62n=   93n+  250n+  337n+  242n+   88n+  125n+   85n
       (5,4,3)    64n+  236n+  231n+  150n+  115n+  125n+    9n=   11n+  161n+  240n+  225n+  106n+  116n+   75n


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