determinant-determinant-x-y-x-2-y-2-586-x-y-x-y-Z-

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

\begin{matrix} \overset{∙}{{\begin{matrix} \underset{x = ?, y = ?}{\overset{x, y \in Z}{x + y + x^{2} y^{2} = 586}} \end{matrix}}_{}^{}} \end{matrix}

Answered by naka3546 last updated on 27/Dec/21

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

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

Also (4, 6), (6, 4)

T han k you sir! B u t t h e p r o c e s s ?

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

x + y + x^{2} y^{2} = 586

{(x y)}^{2} = 586 - (x + y)

\Rightarrow 586 - (x + y) i s p e r f e c t s q u a r e .

586 - (x + y) = a^{2}

x + y = 586 - a^{2}

F o r a = \pm 1, \pm 2, \pm 3, \dots \pm 24

x + y = 586 - 1, 586 - 4, 586 - 9, \dots 586 - 576

\begin{array}{cccccccc} a & x + y & x^{2} y^{2} & x y & z^{2} - (x + y) z + x y = 0 \\ 0 & 586 & 0 & 0 & z^{2} - 586 z = 0 \\ \pm 1 & 585 & 1 & \pm 1 & z^{2} - 585 z \pm 1 = 0 \\ \pm 2 & 582 & 4 & \pm 2 & z^{2} - 582 z \pm 2 = 0 \\ \pm 3 & 577 & 9 & \pm 3 & z^{2} - 577 z \pm 3 = 0 \\ \pm 4 & 570 & 16 & \pm 4 & z^{2} - 570 \pm 4 = 0 \\ \dots & \dots \\ \pm 24 & 10 & 576 & \pm 24 & z^{2} - 10 z \pm 24 = 0 \end{array}

C o n t i n u e \dots