(Note: The exponent $k=3$ has been answered in the affirmative in this post.)
I. Data
For simplicity, assume all terms $\in \mathbb{Z},$ so we can transform the equation to the more symmetric,
$$x_1^5+x_2^5+x_3^5 = y_1^5+y_2^5+y_3^5$$
where $(x_1, x_2, x_3) = (y_1, y_2, y_3)$ are considered trivial solutions. In 2009, by an exhaustive search, Duncan Moore found roughly 5400 primitive solutions within a search radius of about $17700.$ For example,
\begin{align}
1^5 & + 89^5 + 118^5 = 123^5 + 47^5 + 38^5\\
2^5 & + 97^5 + 258^5 = 257^5 + 125^5 + 35^5\\
3^5 & + 54^5 + 62^5 = 67^5 + 28^5 + 24^5\\
4^5 & + 32^5 + 498^5 = 463^5 + 369^5 + 302^5\\
5^5 & + 145^5 + 224^5 = 214^5 + 157^5 + 153^5 \\
6^5 & + 265^5+ 614^5 = 543^5+ 527^5+ 235^5\\
7^5 & + 201^5+ 303^5 = 307^5+ 173^5+ 31^5\\
8^5 & + 62^5+ 68^5 \,=\, 74^5+ 43^5+ 21^5\\
9^5 & + 206^5+ 430^5 = 418^5+ 297^5+ 20^5\\
10^5 & + 100^5+ 972^5 = 951^5+ 617^5+ 204^5\\
\vdots\\[4pt]
200^5 & + 334^5 + 676^5 = 679^5 + 256 ^5 + 185^5
\end{align}
up to $x_1 = 200$ which is quite a long stretch. But there were three missing: namely $x_1 = (22,\,88,\,176)$, all of which are multiples of $11$ and a power of $2$.
Update: The list for $0\leq x_1\leq 1000$ is now complete, with the last two, namely ($410, 840$), found by Oleg567. See his answer below.
II. Updates
- As Adam Bailey pointed out, there doesn't seem to be an obvious congruence obstruction for $x_1 = 22$ and others. So it may be possible for all $x_1$ just like its cousin $z_1^3+z_2^3 = z_4^3+z_4^3$ (though by FLT, this has no $z_1 = 0$).
- There is $0^5 + 220^5 + 14132^5 = 14068^5 + 6237^5 + 5027^5$, so $x_1=0$ is now possible for $5$th powers.
- Oleg567 found $x_1=176$ valid for $k=1,5$ using a larger search radius,
$$176^5+20117^5+22952^5=5781^5+12692^5+24772^5$$
$$176+20117+ 22952=5781+12692+24772$$ - Moore found $x_1=22$ also valid for $k=1,5.$ (See his answer below.)
- Using the form ($5,2,4$), wxffles found $x_1=88$ and $x_1=858$ valid only for $k=5.$ (See addendum to Moore's answer for $x_1<1000$.)
- James Waldby's database for ($5,1,5$) can be found here for other missing $x_1$.
III. Question
Can we in fact find a primitive solution $\in \mathbb{Z}$ for any integer $x_1$,
$$x_1^5+x_2^5+x_3^5 = y_1^5+y_2^5+y_3^5$$
hence the absence of $x_1 = 22, 88, 176,$ etc. is simply an artifact of the search radius?
Best Answer
This is a partial answer. There is a solution to $x_1 =22$, namely,
$$22^5+42303^5+49013^5=51233^5+36563^5+3542^5$$ $$22+42303+49013=51233+36563+3542$$
I. Extended searches
In 2017 I extended my previous exhaustive search (2009) of $x_1^5+x_2^5+x_3^5=y_1^5+y_2^5+y_3^5$ to a radius of $52586$ (i.e. the sum searched to is $52586^5$). There are 15904 solutions and more than half (54.6%) also satisfy $x_1+x_2+x_3=y_1+y_2+y_3$.
In 2017/2018 I ran a separate exhaustive search of the same equation under the constraint $x_1+x_2+x_3=y_1+y_2+y_3\le205848$, giving 16460 solutions, many of which are larger than those in the first set.
The first extended search eliminates 176, as already found by Oleg567. The second search eliminates 22.
II. Statistics
Given $x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k$, statistics for the first extended search are,
$$\begin{array}{|c|c|c|c|c|} \hline \text{Radius} & \text{Solns k = 5} & \text{Solns k = 1,5} & \text{% of k = 1,5} & \text{% diff} \\ \hline 13146 & 3957 & 2378 & 60.1 & --\\ \hline 26293 & 8108 & 4657 & 57.4 & 2.7\\ \hline 39439 & 11982 & 6683 & 55.8 & 1.6\\ \hline 52586 & 15904 & 8677 & 54.6 & 1.2\\ \hline \end{array}$$
To explain, for the first radius of 13146, then $2378/3957 = 0.601$, or 60.1% of solutions (solns) are valid for two exponents $k=1,5$. For the highest radius of 52586, this falls to 54.6%. However, the decline (% diff) seems to be slowing down, so whether this converges to some percentage is unknown.
III. Addendum (by Piezas)
In Moore's ($5,3,3$) database, there are only 9 missing terms with $x_1<1000$: $$88, 275, 410, 495, 785, 800, 840, 850, 858$$ But by adding the databases of wxffles, Waldby, and an identity by Sastry, we find,
$$\color{blue}{88}^5 + 367^5 = 335^5 + 298^5 + 140^5 + 132^5$$ $$\color{blue}{495}^5 + 1043^5 = \color{blue}{858}^5 + 864^5 + 795^5 + 71^5$$ and, $$\color{blue}{275}^5 + 351^5 + 872^5 + 1298^5 + 1855^5 = 1921^5$$ $$\color{blue}{495}^5 + 218^5 + 276^5 + 385^5 + 409^5 = 553^5$$ $$\color{blue}{785}^5 + 1023^5 + 2782^5 + 5407^5 + 5500^5 = 6287^5$$
with the first two from wxffles's ($5,2,4$) database, the last three from James Waldby's ($5,1,5$) database, while Sastry's 1934 parameterization,
$$(\color{blue}{50 n})^5 + (10 n^3)^5 + (n^5 - 25)^5 + (n^5 + 25)^5 = (n^5 - 75)^5 + (n^5 + 75)^5$$
takes care of $x_1$ that are multiples of $50$. Thus only two $x_1<1000$ remain missing: $410, 840$.