(a) Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer. Prove that there exist distinct positive integers
![x, y, z](/media/m/e/1/6/e160f3439547ca8c1afcc35a1c26f080.png)
such that
(b) Let
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
be positive integers such that
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
are relatively prime and
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
is relatively prime either to
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
or to
![b.](/media/m/9/0/a/90a5f50b15448e78bc5971529300acc5.png)
Prove that there exist infinitely many triples
![(x, y, z)](/media/m/f/2/d/f2d4c9b9b3e7f29445f7a1063c15263f.png)
of distinct positive integers
![x, y, z](/media/m/e/1/6/e160f3439547ca8c1afcc35a1c26f080.png)
such that
%V0
(a) Let $n$ be a positive integer. Prove that there exist distinct positive integers $x, y, z$ such that
$$x^{n-1} + y^n = z^{n+1}.$$
(b) Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or to $b.$ Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers $x, y, z$ such that
$$x^a + y^b = z^c.$$