please find the integral solutions (x and y) (xy−7)^2 =x^2 +y^2


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Question Number 31763 by RAMANUJAN last updated on 14/Mar/18

$p l e a s e f i n d t h e i n t e g r a l s o l u t i o n s (x a n d y)$ ${(x y - 7)}^{2} = x^{2} + y^{2}$
	Answered by MJS last updated on 14/Mar/18

	$(- 7, 0)$ $(- 4, - 3)$ $(- 3, - 4)$ $(0, - 7)$ $(0, 7)$ $(3, 4)$ $(4, 3)$ $(7, 0)$
	Commented by MJS last updated on 14/Mar/18

	$. . . now I see that$ $4 \times 3 - 7 = 5$ $4^{2} - 3^{2} = 7$ $3^{2} + 4^{2} = 5^{2}$ $these might have helped with$ $the construction of the given$ $equation$
	Commented by Joel578 last updated on 14/Mar/18

	$Sir pls explain the way . I^{'} m just guessing the$ $answer$
	Commented by MJS last updated on 14/Mar/18

	$I solved the equation$ $y = \frac{7 x \pm \sqrt{x^{4} - x^{2} + 49}}{x^{2} - 1}$ $\Rightarrow x^{4} - x^{2} + 49 must be a square number$ $then I tried . . .$
	Answered by Joel578 last updated on 14/Mar/18

	$(0, \pm 7) and (\pm 7, 0)$
	Answered by ajfour last updated on 14/Mar/18

	$x^{2} y^{2} - 14 x y + 49 = x^{2} + y^{2}$ $x^{2} y^{2} - 16 x y + 64 = {(x - y)}^{2} + 15$ ${(x y - 8)}^{2} = {(x - y)}^{2} + 15$ $(x y - 8 - x + y) (x y - 8 + x - y) = 15$ $\Rightarrow x y - 8 - x + y = 3 a n d$ $x y - 8 + x - y = 5$ $\Rightarrow x y = 12 a n d x - y = 1$ $\Rightarrow (4, 3) o r (- 3, - 4)$ $a l t e r n a t i v e l y$ $x y - 8 - x + y = 5 a n d$ $x y - 8 + x - y = 3$ $\Rightarrow x y = 12 a n d y - x = 1$ $s o (3, 4) o r (- 4, - 3)$ $a l t e r n a t i v e l y$ $x y - 8 - x + y = - 5 a n d$ $x y - 8 + x - y = - 3$ $\Rightarrow x y = 8 a n d x - y = 1$ $\Rightarrow n o i n t e g r a l s o l u t i o n$ $o r x y - 8 - x + y = - 3 a n d$ $x y - 8 + x - y = - 5$ $\Rightarrow x y = 8 a n d y - x = 1$ $\Rightarrow n o i n t e g r a l s o l u t i o n, a g a i n .$ $o t h e r w i s e$ $x y - 8 - x + y = 15 a n d$ $x y - 8 + x - y = 1$ $\Rightarrow x y = 16 a n d y - x = 7$ $\Rightarrow n o i n t e g r a l s o l u t i o n$ $a l t e r n a t i v e l y$ $x y - 8 - x + y = - 15 a n d$ $x y - 8 + x - y = - 1$ $\Rightarrow x y = 0 a n d x - y = 7$ $\Rightarrow (7, 0) a n d (0, - 7)$ $o t h e r w i s e$ $x y - 8 - x + y = - 1 a n d$ $x y - 8 + x - y = - 15$ $\Rightarrow x y = 0 a n d y - x = 7$ $\Rightarrow (0, 7) a n d (- 7, 0)$ $h e n c e t h e i n t e g r a l s o l u t i o n s$ $f o r (x, y) a r e$ $(4, 3), (- 3, - 4), (3, 4), (- 4, - 3),$ $a n d (7, 0), (0, - 7), (0, 7), (- 7, 0) .$
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