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Question Number 70022 by naka3546 last updated on 30/Sep/19

Find   all  pairs  of   (p, q)  integer(s)  such  that  p^3  − q^5   =  (p + q)^2

$${Find}\:\:\:{all}\:\:{pairs}\:\:{of}\:\:\:\left({p},\:{q}\right)\:\:{integer}\left({s}\right)\:\:{such}\:\:{that} \\ $$$${p}^{\mathrm{3}} \:−\:{q}^{\mathrm{5}} \:\:=\:\:\left({p}\:+\:{q}\right)^{\mathrm{2}} \\ $$

Commented by MJS last updated on 30/Sep/19

p=0 q=0  p=0 q=−1  p=1 q=0  p=7 q=3  no other solution for −100≤p, q≤100

$${p}=\mathrm{0}\:{q}=\mathrm{0} \\ $$$${p}=\mathrm{0}\:{q}=−\mathrm{1} \\ $$$${p}=\mathrm{1}\:{q}=\mathrm{0} \\ $$$${p}=\mathrm{7}\:{q}=\mathrm{3} \\ $$$$\mathrm{no}\:\mathrm{other}\:\mathrm{solution}\:\mathrm{for}\:−\mathrm{100}\leqslant{p},\:{q}\leqslant\mathrm{100} \\ $$

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