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Let $\sum_{j=1}^k a_j^n=z^n$. All $a_j,z$ positive integers. $k,n\gt 2$. For a given $n$, for what values of $k$ are there any solutions, and are there only finitely many? For $n=3$, there are solutions for $k=3$. Has this question been studied in detail?

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  • $\begingroup$ For the case k=3 and n=3, is it known whether there are an infinite number of solutions or only a finite number? $\endgroup$ – herb steinberg Mar 23 '18 at 17:53
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This question closely resembles Euler's sum of powers conjecture. Based on the amount of literature on that problem and others inspired by it, I would say the question has been studied in detail. Whether that amount of detail is enough to satisfy your curiosity is another question, but perhaps that keyword gets you started.

Euler's conjecture says, in your notation, that solutions exists only if $k\geq n$, but that is known to be false in general. The unfortunate reality is that the full answer is not known to Euler's conjecture, and therefore your question as its generalization is also in the unknown realm. Some things are known, but there is no complete picture.

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  • $\begingroup$ I looked at the link (Euler's sum..). It looks like only a part of the answer to my question. $\endgroup$ – herb steinberg Mar 22 '18 at 21:46
  • $\begingroup$ @herbsteinberg The conjecture says, in your notation, that solutions exists only if $k\geq n$, but that is known to be false. I agree that it is not your full question, but it is something well studied. Perhaps reading material on generalizations will lead to a fuller question. The unfortunate reality is that the full answer is not known. $\endgroup$ – Joonas Ilmavirta Mar 22 '18 at 22:13

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