1997 IMO 第 6 题

(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 $a c$ is relatively prime to $2 b$. Prove that there exist distinct positive integers $x, y, z$ such that (i) $x^{a}+y^{b}=z^{c}$, and (ii) $x y z$ is relatively prime to $n$.

(a) 设$n$为正整数。证明存在不同的正整数 $x, y, z$ 使得 $$x^{n-1}+y^{n}=z^{n+1} 。$$ (b) 设$a、b、c$ 为正整数，使得$a$ 和$b$ 互质，并且$c$ 与$a$ 或$b$ 互质。证明存在无限多个不同正整数 $x, y, z$ 的三元组 $(x, y, z)$ ，使得 $$x^{a}+y^{b}=z^{c} 。$$ 原始公式：设$a、b、c、n$ 为正整数，使得$n$ 为奇数且$a c$ 与$2 b$ 互质。证明存在不同的正整数 $x, y, z$，使得 (i) $x^{a}+y^{b}=z^{c}$，并且 (ii) $x y z$ 与 $n$ 互质。