While answering this question, I decided to revisit the equation,
$$x_1^6+x_2^6+x_3^6+\dots+x_7^6 = z^6$$
which I asked about in an old post and, with today's faster computers, might yield some new insights. Primitive solutions to the equation,
$$x_1^k+x_2^k+x_3^k+\dots+x_{k+1}^k = z^k$$
where k+1 is a prime p come in two kinds: 1st, $z$ is integrally divisible by $p$, or 2nd, it is not. The 1st kind has smaller solutions than the 2nd kind, but it is the latter we are interested in since there is a possibility one of its terms is $x_i=0$.
I. 4th powers: $(k+1 = 5)$
$$2^4 + 2^4 + 3^4 + 4^4 + 4^4 = 5^4$$ $$10^4 + 10^4 + 10^4 + 17^4 + 30^4 = 31^4 = z_1^4$$ $$\;\color{red}{0^4} + 30^4 + 120^4 + 272^4 + 315^4 = 353^4 = z_2^4$$
Note that for the 2nd kind, then $z_2=353$ is more than 10x the smaller solution $z_1=31$ (which is prime). This may give us a rough idea of what to expect for 6th powers.
II. 6th powers: $(k+1 = 7)$
$$74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6 = 1141^6 =(7\times163)^6$$ $$42^6(195^6 + 260^6 + 440^6 + 506^6 + 580^6) + 19229^6 + 33354^6 = 34781^6$$ $$\color{red}{0^6}+x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6 = z_2^6$$
Based on 4th powers, we could assume that $z_2$ is more than 14x the smaller solution $z_1=34781$ (which is also prime), or roughly $z_2 > 480000$. In fact, according to a 2002 paper, Meyrignac claims no solutions with $z_2 < 730000$.
Edit: I just noticed that $730000/34781 \approx 20.99$, so it seems $z_2$ is more than 21x the smaller solution $z_1$. If the trend continues, it does not bode well for $10$th powers.
III. Tables for 6th powers
In Meyrignac's site http://euler.free.fr/database.txt, there are exactly $100+78=178$ known solutions to the equation (6,1,7) with $z<412000$. We are interested in the $78$ solutions of the 2nd kind. Statistics are,
$$\begin{array}{|c|c|c|c|} \hline \text{Range name}&\text{Range of z}&\text{Solutions}&\text{Year}\\ \hline R_1& \qquad 1-100000& 12 &2000\\ \hline R_2& 100001-200000& 17&2000\\ \hline R_3& 200001-300000& 32&2006-2008\\ \hline R_{4a}& 300001-350000& 16&2008-2012\\ \hline R_{4b}& 350001-400000& 0&\text{(gap)}\\ \hline R_5& 400001-500000& 1&2010\\ \hline \text{Total}& \;\qquad 1-500000 &78& 2000-2012^{\large{*}}\\ \hline \end{array}$$
Large $z$ with $z<350000$ were found mostly by Rolan Christofferson between 2006-2012. But recently in 2021, Ian Stopher found an additional solution in that range, namely $z=319799$. So obviously it was not an exhaustive search.
IV. Questions
Assume solutions of the 2nd kind. The first twelve known are $R_1$ = (34781, 38191, 39779, 54347, 59819, 63631, 66029, 67681, 70513, 78919, 89797, 97081).
- Do we now have the computing power to do an exhaustive search for $z<200000$? (It is expected $R_1 = 12,\,R_2 = 19$ will be sparse, but they seem too sparse compared to $R_3= 32$. And $R_1$ doesn't seem to be evenly spread with nothing for 40000-50000.)
- Or power even for $z<400000$? Since $R_{4a} = 16$, then $R_{4b}$ may be similar and the total could be $R_4 > (R_3 = 32)$.
