We can observe the Armstrong numbers. Similarly, there exists numbers like $166 \cdots 66^3 + 500 \cdots 00^3 + 33 \cdots 33^3 = 16\cdots66\cdots500 \cdots 00 \cdots 33 \cdots 33$. A lot of such patterns can be found here.
I have answers to the $166..^3+500..^3+33..33^3=16..6500..0033..3$ and the finiteness of Armstrong numbers. Anything else?
But, is there any generalisation on any of these? Like, can we say if there exists infinitely many numbers $a,b,c$ such that $a^3+b^3+c^3 = \overline{abc}$. ($a,b,c$ are all $\geq 2$-digit numbers and $\overline {abc}$ represents the decimal representation)? Or maybe, a more generalised version, if exists, like : $x_1^k + x_2^k + \cdots + x_n^k = \overline {x_1x_2 \cdots x_n}$?
Is there any paper (on arxiv or something else) that I can find interesting and are somewhat like these patterns? The patterns look interesting actually. Anything related to this (which maybe even a distant apart but still has some mere relation) will also be interesting to learn about.
And, please don't stop posting answers to this. I am interested to know about these, so, just carry on posting answers. I don't think there can be a definite answer (post) to this- so, carry on increasing the number of answers to this. I'm just weirdly interested to know how far this might go!