Question Number 211753 by alcohol last updated on 19/Sep/24
$${find}\:{all}\:\left({n},{m}\right)\:{such}\:{that}\:\frac{{n}^{\mathrm{2}} −{m}}{{m}^{\mathrm{2}} −{n}}\:\in\:\mathbb{Z} \\ $$
Answered by Frix last updated on 20/Sep/24
$$\mathrm{There}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many}\:\mathrm{solutions}. \\ $$$$\mathrm{3}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{find}\:\mathrm{some}: \\ $$$$\mathrm{1}.\:\mathrm{Let}\:{m}\in\mathbb{Z}\backslash\left\{\mathrm{0},\:\mathrm{1}\right\}\wedge{n}={m}\:\Rightarrow\:\frac{{n}^{\mathrm{2}} −{m}}{{m}^{\mathrm{2}} −{n}}=\mathrm{1} \\ $$$$\mathrm{2}.\:\mathrm{Let}\:{m}\in\mathbb{Z}\wedge{n}=−\left({m}+\mathrm{1}\right)\:\Rightarrow\:\frac{{n}^{\mathrm{2}} −{m}}{{m}^{\mathrm{2}} −{n}}=\mathrm{1} \\ $$$$\mathrm{3}.\:\mathrm{Let}\:{m}\in\mathbb{Z}\wedge{n}=−{m}\left({m}^{\mathrm{3}} −{m}−\mathrm{1}\right)\:\Rightarrow \\ $$$$\:\:\:\:\:\frac{{n}^{\mathrm{2}} −{m}}{{m}^{\mathrm{2}} −{n}}={m}^{\mathrm{4}} −\mathrm{2}{m}^{\mathrm{2}} −{m}+\mathrm{1} \\ $$