Question 6
(IRE) (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
$$x^a+y^b=z^c.$$
Original formulation: Let $a,b,c,n$ be positive integers such that $n$ is odd and $ac$ is relatively prime to $2b$. Prove that there exist distinct positive integers $x,y,z$ such that
(i) $x^a+y^b=z^c$, and
(ii) $xyz$ is relatively prime to $n$.
$$x^{n-1}+y^n=z^{n+1}.$$
(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
$$x^a+y^b=z^c.$$
Original formulation: Let $a,b,c,n$ be positive integers such that $n$ is odd and $ac$ is relatively prime to $2b$. Prove that there exist distinct positive integers $x,y,z$ such that
(i) $x^a+y^b=z^c$, and
(ii) $xyz$ is relatively prime to $n$.