1.Introduction


I will show that x1^k+x2^k+x3^k+x4^k+x5^k+x6^k+x7^k+x8^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k+y7^k+y8^k   
for k=1,3,5,7,9 has infinetly many integer solutions.

I used following Letac-Sinha identity and Theorem.

Letac-Sinha identity.([1])

(a+c)^k + (a-c)^k + (3b+d)^k + (3b-d)^k + (4a)^k = (3a+c)^k + (3a-c)^k + (b+d)^k + (b-d)^k + (4b)^k,
for k=1,2,4,6,8,where a^2+12b^2=c^2 and 12a^2+b^2=d^2.


Theorem

If a1^k+a2^k+....+am^k = b1^k+b2^k+....+bm^k,for k=2,4,...2n,then

(T+a1)^k+(T+a2)^k+...+(T+am)^k,(T-a1)^k+(T-a2)^k+...+(T-am)^k =

(T+b1)^k+(T+b2)^k+...+(T+bm)^k,(T-b1)^k+(T-b2)^k+...+(T-bm)^k,
for k=1,2,3,...2n+1,where T is arbitrary integer.









2.x1^k+x2^k+x3^k+x4^k+x5^k+x6^k+x7^k+x8^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k+y7^k+y8^k
  for k=1,3,5,7,9 has infinetly many solutions,
  where a^2+12b^2=c^2 and 12a^2+b^2=d^2.
         
     
     
Proof.

x1^k+x2^k+x3^k+x4^k+x5^k+x6^k+x7^k+x8^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k+y7^k+y8^k................(1)

Let [a1,a2,...,am]=[b1,b2,...,bm](k=1,2,...,n) denote a1^k+a2^k+....+am^k = b1^k+b2^k+....+bm^k(k=1,2,...,n).

First we apply Letac-Sinha identity to above Theorem,we obtain 


[T+a+c, T+a-c, T+3b+d, T+3b-d, T+4a, T-a-c, T-a+c, T-3b-d, T-3b+d, T-4a]^k =

[T+3a+c, T+3a-c, T+b+d, T+b-d, T+4b, T-3a-c, T-3a+c, T-b-d, T-b+d, T-4b]^k,for k=1,3,5,7,9,
where a^2+12b^2=c^2 and 12a^2+b^2=d^2.

Here,set T=-c,we can reduce four terms and obtain


[-2c+a, -c+3b+d, -c+3b-d, -c+4a, -2c-a, -c-3b-d, -c-3b+d, -c-4a]^k =

[-2c+3a, -c+b+d, -c+b-d, -c+4b, -2c-3a, -c-b-d, -c-b+d, -c-4b]^k,for k=1,3,5,7,9,

where a^2+12b^2=c^2 and 12a^2+b^2=d^2.


Since Letac-Sinha identity has infinetly many integer solutions,
this identity also has infinetly many integer solutions.


Q.E.D.　
 
　　                  
       
3.Small Solutions?

[       a    ,    b  ,      c     ,      d     ]  [   x1 ,   x2 ,   x3  ,   x4  ,   x5  ,   x6  ,  x7  ,  x8  ]
                                                  [   y1 ,   y2 ,   y3  ,   y4  ,   y5  ,   y6  ,  y7  ,  y8  ]

[-11869/11881, -2/109, 11893/11881, 41116/11881]  [-35655, 28569, -53663, -59369, -11917, -52355, 29877, 35583]
                                                  [-59393, 29005, -53227, -12765,  11821, -52791, 29441,-11021]

[-218/20667, -143/249, 41116/20667, 11893/20667]  [-41225, -32415, -44308, -20994, -41007,  -8701, 3192,-20122]
                                                  [-41443, -20546, -32439, -44296, -40789, -20570,-8677,  3180]


 






4.References

[1].Tito Piezas:http://sites.google.com/site/tpiezas/028