Question Number 151316 by iloveisrael last updated on 20/Aug/21
$$\:\:\:\:\:\:{Find}\:{positive}\:{integers}\: \\ $$$$\:\:\:\:\:{a}\:{and}\:{b}\:{such}\:{that}\: \\ $$$$\:\:\:\:\:\left(\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:−\mathrm{1}\right)^{\mathrm{2}} =\:\mathrm{49}+\:\mathrm{20}\sqrt[{\mathrm{3}}]{\mathrm{6}}\: \\ $$
Answered by Rasheed.Sindhi last updated on 20/Aug/21
$$\overset{\mathcal{SOLVE}\:\:{for}\:{a},{b}\in\mathbb{Z}^{+} } {\left(\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:−\mathrm{1}\right)^{\mathrm{2}} }=\:\mathrm{49}+\:\mathrm{20}\sqrt[{\mathrm{3}}]{\mathrm{6}}\: \\ $$$$\left(\underset{{m}} {\underbrace{\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:}}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{49}=\mathrm{20}\sqrt[{\mathrm{3}}]{\mathrm{6}}\: \\ $$$$\Rightarrow{m}\:{is}\:{not}\:{rational} \\ $$$$\left({m}\:−\mathrm{1}−\mathrm{7}\right)\left({m}\:−\mathrm{1}+\mathrm{7}\right)=\mathrm{20}\sqrt[{\mathrm{3}}]{\mathrm{6}} \\ $$$$\left({m}−\mathrm{8}\right)\left({m}\:+\mathrm{6}\right)=\mathrm{1}.\mathrm{2}^{\mathrm{2}} .\mathrm{5}.\sqrt[{\mathrm{3}}]{\mathrm{2}}.\sqrt[{\mathrm{3}}]{\mathrm{3}} \\ $$$$\left({Obviously},\:{m}\:−\mathrm{8}<{m}+\mathrm{6}\right) \\ $$$${perhaps}\:{no}\:{solution} \\ $$