1986 IMO 第 5 题

(FRG 1) ${ }^{\mathrm{IMO1}}$ The set $S=\{2,5,13\}$ has the property that for every $a, b \in S, a \neq b$, the number $a b-1$ is a perfect square. Show that for every positive integer $d$ not in $S$, the set $S \cup\{d\}$ does not have the above property.

(FRG 1) ${ }^{\mathrm{IMO1}}$ 集合 $S=\{2,5,13\}$ 具有以下性质：对于每个 $a, b \in S, a \neq b$，数字 $a b-1$ 是完全平方数。证明对于不在 $S$ 中的每个正整数 $d$，集合 $S \cup\{d\}$ 不具有上述性质。