We know that $a^2 + b^2 = c^2$ has infinitely integer solutions which can be written as 3-tuples for example (3,4,5).

Can we conjecture that $$x_{1}^n +x_{2}^n + \cdots +x_{n}^n = y^n$$ has infinitely many integer solutions for each $n \in \mathbb N$? If so how would one prove this to be correct?

I apologize if this is a simple question. I havent taken number theory but this is something that has caught my curiosity. Thanks in advance.

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    $\begingroup$ related: mathoverflow.net/questions/62820/pythagorean-5-tuples $\endgroup$ – Michael McGovern Jul 26 '18 at 22:52
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    $\begingroup$ An answer to this would follow from solutions to Waring's problem. $\endgroup$ – Somos Jul 27 '18 at 0:00
  • $\begingroup$ Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers. $\endgroup$ – Gerry Myerson Jul 27 '18 at 7:03
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    $\begingroup$ But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=\cdots=x_n=0$. $\endgroup$ – Gerry Myerson Jul 27 '18 at 7:04
  • $\begingroup$ According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers." $\endgroup$ – Gerry Myerson Jul 27 '18 at 7:07

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