Why do the numbers in the generalization of Pythagorean triples have approximately the same size?

Question

Below is a list of partitions of perfect powers into numbers of the same power. Some of them are counterexamples to Euler's sum of powers conjecture, and the others are generalizations of Pythagorean triples.

What strikes me is that numbers on the left hand side seem to be (in some sense) of approximately the same size as the number on the left hand side. There are exceptions, agreed, for example the $7^5$ and $55^5$ terms below. But, for example, patterns of the type $1^7+3^7+6^7+9^7+432^7+1234^7+14511^7=\;?^7$ seem to be lacking.

The $k=2$ case (Pythagorean triples) can probably be explained quite easily with elementary methods, like the non-rigorous reasoning:

$(m+n)^2$ where n is small is much bigger than $m^2$, which requires a quite large $a^2$ to form the triple $a^2+m^2=(m+n)^2$.

But when $k \geq 4$, there appear to be more "degrees of freedom" to possibly allow smaller terms.

Is the observation for $k\geq 4$ true in general and, in that case, is there an explanation to it?

"Euler" partitions into powers

$$\begin{align} 30^4 + 120^4 + 272^4 + 315^4 &= 353^4\\ 95800^4+217519^4+414560^4&=422481^4\\ 2682440^4+15365639^4+18796760^4&=20615673^4\\\\ 7^5 + 43^5 + 57^5 + 80^5 + 100^5 &= 107^5\\ 19^5 + 43^5 + 46^5 + 47^5 + 67^5 &= 72^5\\ 27^5+84^5+110^5+133^5&=144^5\\ 55^5+3183^5+28969^5+85282^5&=85359^5\\\\ 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 &= 568^7\\\\ 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8 &= 1409^8 \end{align} $$

Answer 1

For more data, look at these databases: Wroblewski's for $(4,1,4)$ and Waldby's for $(5,1,5)$.

$\color{blue}{(k=4)}$: Let us quantify what you have observed. You say, "What strikes me is that numbers on the left hand side seem to be (in some sense) of approximately the same size..." I assume you are basing this on the smallest soln to $x_1^4+x_2^4+x_3^4+x_4^4 = z^4$ with $x_1<x_2<x_3<x_4$ as,

$$30^4 + 120^4 + 272^4 + 315^4 = 353^4\tag1$$

Let us interpret the vague phrase "in some sense" as looking for other solutions which have adjacent addends with similar "ratios" to $(1)$, namely (rounded to the first decimal place),

$$x_4/x_3 = 315/272 < 1.2,\quad x_3/x_2 = 272/120 < 2.3$$

Using Wroblewski's database of $1009$ solutions, Mathematica counts there are only $241$ with those bounds, so $241/1009 \approx 0.238$, or less than $24\text{%}$. Thus, it seems the phenomenon you have observed is only in a minority.

But if we are generous and use the nearest integer ceiling,

$$x_4/x_3 < 2,\quad\quad x_3/x_2 < 3$$

then Mathematica counts $729$ solutions, so $729/1009 \approx 0.722$, or about $\color{red}{72\text{%}}$, and is now in the majority. Thus, it depends on the bounds that one uses. However, one can argue that integers are a more "natural" choice.

$\color{blue}{(k=5)}$: The smallest is,

$$19^5 + 43^5 + 46^5 + 47^5 + 67^5 = 72^5$$

with the ratios (rounded to the first decimal place ceiling),

$$x_5/x_4 = 67/47 < 1.5,\quad x_4/x_3 = 47/46 < 1.1,\quad x_3/x_2 = 46/43 < 1.1$$

In Waldby's database of $410$ solutions, there are only $13$ with those bounds, so $13/410 \approx 0.03$, or just $3\text{%}$. However, if we use ratios in a integer linear progression like the second case for $k=4$,

$$x_5/x_4 < 2,\quad x_4/x_3 < 3,\quad x_3/x_2 < 4$$

then there are $326$, so $326/410 \approx 0.795$, or almost $\color{red}{80\text{%}}$.

Caveat: The phenomenon you observed is highly sensitive to the bounds that one uses, and it can be in a minority or a majority (or anything in between).

$\color{blue}{\text{(Exceptions)}}$: Recall that we have,

$$\color{brown}{1^3} + (9n^4 - 3n )^3 + (9n^3 - 1)^3 = (9n^4)^3$$

You remark, "When $k\geq4$ , there appear to be more "degrees of freedom" to possibly allow smaller terms..."

And yes there is.

$$\color{brown}{2^5 + 5^5} + 147^5 + 1910^5 + 3803^5 = 3827^5$$

$$\color{brown}{8^5+ 8^{10}}+ 2859^5+3026^5+ 3636^5 = 4043^5$$

When $k+1$ is prime, there are congruence restrictions on the addends (see this post) and which may inhibit multiple small terms. For $k=7$, judging from the "trend", the pattern,

$$\color{brown}{x_1^7+ x_2^7+x_3^7}+ x_4^7 +x_5^7 +x_6^7 +x_7^7 = z^7$$

where the three brown $x_i$ are relatively small to the other addends may be feasible. Unfortunately, there are only three known solutions in positive integers, so we will have to wait and see until many more are found.