(a) Let
be a positive integer. Prove that there exist distinct positive integers
such that
(b) Let
be positive integers such that
and
are relatively prime and
is relatively prime either to
or to
Prove that there exist infinitely many triples
of distinct positive integers
such that
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(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
$$x^a + y^b = z^c.$$
Školjka