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It is well known that for all $n>2$, there is no integer solution for the following:

$$x^n + y^n = z^n $$

I will modify this equation slightly. What if the number of terms on the left-hand side is equal to $n$? For instance $n=5$:

$$a_{1}^{5}+a_{2}^{5}+a_{3}^{5}+a_{4}^{5}+a_{5}^{5}=+a_{6}^{5}$$

Is there solution for such equations, where the $a_k$'s are integer? (I mean non-trivial)

Severus' Constant
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  • There is always a solution with $a_i=0$, for $i=1,\ldots, n-1$ and $a_{n}=a_{n+1}$. These are trivial solutions. So there are integer solutions for $x^n+y^n=z^n$ for all $n>2$. – Dietrich Burde Nov 27 '23 at 10:48

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We have the following non-trivial solution $$ 72^5=19^5+43^5+46^5+47^5+67^5. $$ Here $72$ is the smallest positive integer with this property. For more examples see here. In general, Euler's conjecture is about your equation. It was disproved for $n=4,5$ but is still open for $n\ge 6$.

Dietrich Burde
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    I marked this as accepted answer. Because what I am looking for was whether the conjecture exists. – Severus' Constant Nov 27 '23 at 10:57
  • In the simplest FLT case with power three you have only n-1 nth powers adding up to another one. Here you have n numbers. So you have c n^5 sums less than n^5. – gnasher729 Nov 27 '23 at 11:50