The Diophantine equation,

$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$ 

for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of **primitive** solutions (as points on an *elliptic curve*, or an infinite family of polynomials,). However,

$$x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k\tag2$$

for either $k=7$ or $8$, results seem to be harder to find. Only ***one*** solution (found back in 2006) is known for $k=8$. Many solutions exist when $k=7$, but it is not known if there are infinite many. Choudhry [reduced][1] $(2)$ when it is true for $k=1,3,7$ to a multi-variable *cubic equation* by passing through the simultaneous equations,

$$\begin{aligned}
abc\,&=def\\
2(a^4 + b^4 + c^4) - 5(a^2 + b^2 + c^2)^2\, &= 2(d^4 + e^4 + f^4) - 5(d^2 + e^2 + f^2)^2
\end{aligned}\tag3$$

Using that cubic, Choudhry (3) and Wroblewski (21) found a total of $3+21=24$ solutions to $(2)$ valid for $k=1,3,7$. Results can be found in [euler.free.fr][2]. Using the $x_i,y_i$, I derived their $a,b,c,d,e,f$. Surprisingly, seven had a simple linear constraint, namely $c=2f$. Let,

$$a,\,b,\,c =\alpha\beta,\;\gamma\lambda,\;2t$$
$$d,\,e,\,f =2\alpha\lambda,\;\beta\gamma,\;t$$

then solutions are,

$$\begin{array}{|c|c|c|c|c|c|}
\text{#}&\alpha&\beta&\gamma&\lambda&t\\
1&227 &27 &113 &13 &1305\\
2&431 &187 &365 &49 &6929\\
3&127 &139 &313 &1 &23647\\
4&303 &304 &338 &37 &24616\\
5&871 &163 &4364 &127 &254405 \\
6&439 &2459 &247 &1175 &261851 \\
7&14737 &139 &15899 &25 &544069 \\
\end{array}$$

It was hard to find commonalities for the other seventeen solutions.

**Questions:**

1. Can the simultaneous eqns $(3)$, or the cubic described by Choudhry in the paper, be reduced to an *elliptic curve*? 
2. Why does almost $1/3$ of known solutions have $c=2f$?

**P.S.** Eq.1 is briefly discussed in this [MO post][3].


  [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.101.2751
  [2]: http://euler.free.fr/database.txt
  [3]: http://mathoverflow.net/questions/150428/