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Question Number 162169 by Rasheed.Sindhi last updated on 27/Dec/21

determinant ((( determinant ((( x+y+x^2 y^2 =586_(x=?,y=? ) ^(x,y∈Z ) )))_ ^ _() ^(•) )))

$$ \\ $$$$\:\:\:\:\:\:\:\begin{array}{|c|}{\overset{\bullet} {\:\:\:\:\:\begin{array}{|c|}{\:\:\:\underset{{x}=?,{y}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {\overset{{x},{y}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {{x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586}}}\:\:}\\\hline\end{array}_{} ^{} }\:\:\:\:}\\\hline\end{array} \\ $$$$ \\ $$

Answered by naka3546 last updated on 27/Dec/21

(x,y) = (−2,12) , (0, 586) , (12, −2) , (586, 0)

$$\left({x},{y}\right)\:=\:\left(−\mathrm{2},\mathrm{12}\right)\:,\:\left(\mathrm{0},\:\mathrm{586}\right)\:,\:\left(\mathrm{12},\:−\mathrm{2}\right)\:,\:\left(\mathrm{586},\:\mathrm{0}\right) \\ $$

Commented by Rasheed.Sindhi last updated on 27/Dec/21

Also (4,6),(6,4) Thank you sir!But the process?

$$\mathrm{Also}\:\left(\mathrm{4},\mathrm{6}\right),\left(\mathrm{6},\mathrm{4}\right) \\ $$$$\mathbb{T}\mathrm{han}\Bbbk\:\mathrm{you}\:\mathrm{sir}!{But}\:{the}\:{process}? \\ $$

Answered by Rasheed.Sindhi last updated on 28/Dec/21

x+y+x^2 y^2 =586 (xy)^2 =586−(x+y) ⇒586−(x+y) is perfect square. 586−(x+y)=a^2 x+y=586−a^2 For a=±1,±2,±3,...±24 x+y=586−1,586−4,586−9,...586−576 determinant ((a,(x+y),(x^2 y^2 ),(xy),(z^2 −(x+y)z+xy=0)),(0,(586),( 0),0,(z^2 −586z=0)),((±1),(585),1,(±1),(z^2 −585z±1=0)),((±2),(582),4,(±2),(z^2 −582z±2=0)),((±3),(577),9,(±3),(z^2 −577z±3=0)),((±4),(570),(16),(±4),(z^2 −570±4=0)),((...),(...),,,),((±24),(10),(576),(±24),(z^2 −10z±24=0))) Continue...

$${x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586} \\ $$$$\left({xy}\right)^{\mathrm{2}} =\mathrm{586}−\left({x}+{y}\right)\: \\ $$$$\Rightarrow\mathrm{586}−\left({x}+{y}\right)\:{is}\:{perfect}\:{square}. \\ $$$$\:\:\:\:\mathrm{586}−\left({x}+{y}\right)={a}^{\mathrm{2}} \\ $$$$\:\:\:\:{x}+{y}=\mathrm{586}−{a}^{\mathrm{2}} \\ $$$${For}\:{a}=\pm\mathrm{1},\pm\mathrm{2},\pm\mathrm{3},...\pm\mathrm{24} \\ $$$${x}+{y}=\mathrm{586}−\mathrm{1},\mathrm{586}−\mathrm{4},\mathrm{586}−\mathrm{9},...\mathrm{586}−\mathrm{576} \\ $$$$\begin{array}{|c|c|c|c|c|c|c|c|}{{a}}&\hline{{x}+{y}}&\hline{{x}^{\mathrm{2}} {y}^{\mathrm{2}} }&\hline{{xy}}&\hline{{z}^{\mathrm{2}} −\left({x}+{y}\right){z}+{xy}=\mathrm{0}}\\{\mathrm{0}}&\hline{\mathrm{586}}&\hline{\:\:\mathrm{0}}&\hline{\mathrm{0}}&\hline{{z}^{\mathrm{2}} −\mathrm{586}{z}=\mathrm{0}}\\{\pm\mathrm{1}}&\hline{\mathrm{585}}&\hline{\mathrm{1}}&\hline{\pm\mathrm{1}}&\hline{{z}^{\mathrm{2}} −\mathrm{585}{z}\pm\mathrm{1}=\mathrm{0}}\\{\pm\mathrm{2}}&\hline{\mathrm{582}}&\hline{\mathrm{4}}&\hline{\pm\mathrm{2}}&\hline{{z}^{\mathrm{2}} −\mathrm{582}{z}\pm\mathrm{2}=\mathrm{0}}\\{\pm\mathrm{3}}&\hline{\mathrm{577}}&\hline{\mathrm{9}}&\hline{\pm\mathrm{3}}&\hline{{z}^{\mathrm{2}} −\mathrm{577}{z}\pm\mathrm{3}=\mathrm{0}}\\{\pm\mathrm{4}}&\hline{\mathrm{570}}&\hline{\mathrm{16}}&\hline{\pm\mathrm{4}}&\hline{{z}^{\mathrm{2}} −\mathrm{570}\pm\mathrm{4}=\mathrm{0}}\\{...}&\hline{...}&\hline{}&\hline{}&\hline{}\\{\pm\mathrm{24}}&\hline{\mathrm{10}}&\hline{\mathrm{576}}&\hline{\pm\mathrm{24}}&\hline{{z}^{\mathrm{2}} −\mathrm{10}{z}\pm\mathrm{24}=\mathrm{0}}\\\hline\end{array} \\ $$$${Continue}... \\ $$

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