Given the Diophantine equation,

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

there is rather the curious observation that the smallest positive solutions for $k=5$ or $6$ is multi-grade.

$$24^k+28^k+67^k=3^k+54^k+62^k,\quad k = 1,5$$

$$15^k + 10^k + 23^k = 3^k + 19^k + 22^k,\quad k = 2,6$$

Duncan Moore has exhaustively searched $(1)$ for all positive solutions below a bound. [Table 1][1] is for $k=5$, while [Table 2][2] is for $k=6$. We can summarize the data below.

I. Table 1:

$$\begin{array}{|c|c|c||} \text{# of solns}&\color{blue}{A=\text{% of}\; (k = 1,5)}&\text{diff}\\ 168&63.7\text{%}& \\ 336&65.8\text{%}&+2.7\\ 672&65.6\text{%}&-0.3 \\ 1344&63.6\text{%}&-2.0\\ 2688&61.0\text{%}&-2.6\\ 5376&59.1\text{%}&-1.9\\ \end{array}$$

Note: Each row doubles the $\text{#}$. Since Moore's database has $5393$ solns, and $5393/2^5\approx168$, then that's where I started.

II. Table 2:

$$\begin{array}{|c|c|c|} \text{# of solns}&\color{blue}{B=\text{% of}\; (k = 2,6)}&\text{diff}\\ 50&80\text{%}& \\ 100&85\text{%}&+5.0\\ 200&89\text{%}&+4.0\\ 400&91.7\text{%}&+2.7\\ \end{array}$$

Questions:

1. Why is the percentage of $A$ decreasing, while that of $B$ is apparently increasing? Or will $B$ eventually have a negative diff like $A$? (The data is too small to be conclusive.)
2. If both are decreasing, will $A,B \to 0$? Or will it taper off to some constant?

P.S. Similarly, the smallest solutions to,

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

are also multigrades as $k=1,5$, and $k=2,6$, though there are no exhaustive tables for these. [1]: http://homepage.ntlworld.com/duncan-moore/taxicab/3_3_5.txt [2]: http://homepage.ntlworld.com/duncan-moore/taxicab/3_3_6.txt