Question Number 222117 by klipto last updated on 18/Jun/25
$$\boldsymbol{\mathrm{problem}}\:\mathrm{1}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integer}} \\ $$$$\boldsymbol{\mathrm{m}},\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{triples}}\left(\boldsymbol{\mathrm{n}},\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\right)\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}},\boldsymbol{\mathrm{with}} \\ $$$$\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{relatively}}\:\boldsymbol{\mathrm{prime}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{m}},\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{satisfy}} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} \right)^{\boldsymbol{\mathrm{m}}} =\left(\boldsymbol{\mathrm{xy}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$$$\boldsymbol{\mathrm{hint}}:\boldsymbol{\mathrm{utilize}}\:\boldsymbol{\mathrm{AM\&GM}},\boldsymbol{\mathrm{diophantine}}\:\boldsymbol{\mathrm{eqn}} \\ $$$$\boldsymbol{\mathrm{KLIPTO}}−\boldsymbol{\mathrm{QUANTA}}−\boldsymbol{\mathrm{OOZY}} \\ $$