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.