Question Number 188442 by BaliramKumar last updated on 03/Mar/23
$$ \\ $$ $${Solve}\:{by}\:{computer}\:{programing} \\ $$ $$\left({if}\:{possible}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{d}<{a}<{b}\:\&\:{c}\:<\:{a},\:{b}>\mathrm{2}{c} \\ $$ $$\cancel{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\:\mathrm{5}{c}^{\mathrm{2}} +\mathrm{2}{d}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\left({a},\:{b},\:{c},\:{d}\:\:\in\:\mathrm{N}\right) \\ $$ $${c}^{\mathrm{2}} +{d}^{\mathrm{2}} \:=\:{a}^{\mathrm{2}} \:\:\:\:\:\:\:\:.........\left({i}\right) \\ $$ $$\left(\mathrm{2}{c}\right)^{\mathrm{2}} +{d}^{\mathrm{2}} \:=\:{b}^{\mathrm{2}} \:\:.........\left({ii}\right) \\ $$ $$ \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a},\:{b},\:{c},\:{d}\right)\:=\:? \\ $$
Answered by nikif99 last updated on 02/Mar/23
$${Many}\:{solutions}\:\left(\mathrm{26}\:{for}\:{a},\:{b},\:{c},\:{d}\:\leqslant\mathrm{30}\right) \\ $$
Commented bynikif99 last updated on 02/Mar/23
Commented byBaliramKumar last updated on 03/Mar/23
$${now}\:\mathrm{2}\:{equations\&}..\downdownarrows \\ $$ $${a}>{d} \\ $$ $${a}>{c} \\ $$ $${b}>{a} \\ $$ $${b}>\mathrm{2}{c} \\ $$ $${b}>{d} \\ $$
Commented byBaliramKumar last updated on 03/Mar/23
Commented bynikif99 last updated on 03/Mar/23
$${No}\:{integer}\:{solution}\:{for}\:{a},\:{b},\:{c},\:{d}\:\leqslant\mathrm{1000} \\ $$
Commented byBaliramKumar last updated on 03/Mar/23
$${Thanks}\:{Sir}\:\:\:\:\:\:{for}\:{information} \\ $$