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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!

Mathejunior
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  • Related : How to prove this curiosity that has to do with cubes of certain numbers? – mathlove Mar 01 '18 at 06:53
  • @mathlove I know that. That's a pretty famous thing. No need to delete your comment though but i have seen it a lot of times before. Anything else (apart from the 166...500...333... thing) – Mathejunior Mar 01 '18 at 06:57
  • I think that the link is the answer to "can we say if there exists infinitely many numbers $a,b,c$ such that $a^3+b^3+c^3 = \overline{abc}$." It seems that you might want to make your question clearer. – mathlove Mar 01 '18 at 07:10
  • Check it now. It's been edited. – Mathejunior Mar 01 '18 at 09:44
  • Well, your question is unclear about whether you count $166..^3+500..^3+33..33^3=16..6500..0033..3$ as one example or as inifinitely many examples where $1^3+5^3+3^3=153,16^3+50^3+33^3=165033$ and so on. – mathlove Mar 01 '18 at 10:21
  • @mathlove well, it's same I guess. – Mathejunior Mar 01 '18 at 10:25
  • It matters. If you count $333..3^3+666..7^3+000..0^3=333..3666..7000..0$ as infinitely many examples where $3^3+7^3+0^3=370,33^3+67^3+00^3=336700$ and so on, then one of your questions "can we say if there exists infinitely many numbers $a,b,c$ such that $a^3+b^3+c^3 = \overline{abc}$" is solved. – mathlove Mar 01 '18 at 10:39
  • @mathlove Yeah, that's another. Nah, that's an example to my question. Btw, there's nothing like "solving the question". I am just trying to find as general solution as possible. Anything related will also do. Basically I would be more interested into something analytical than just examples. – Mathejunior Mar 01 '18 at 10:49
  • oeis-A056733 where each number is the sum of the cubes of its 3 sections. – mathlove Mar 01 '18 at 10:59

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