solve in Z x^3 +6y^3 =4z^3


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Question Number 35291 by 123 45 polytechnicien last updated on 17/May/18

$s o l v e i n Z x^{3} + 6 y^{3} = 4 z^{3}$
Commented by 123 45 polytechnicien last updated on 17/May/18

$w h y z = x + i y a n d n o t a + i b ?$
	Answered by ajfour last updated on 17/May/18

	$4 {(x + i y)}^{3} = x^{3} + 6 y^{3}$ $4 (x^{3} - {i y}^{3} + 3 {i x}^{2} y - 3 {x y}^{2}) = x^{3} + 6 y^{3}$ $\Rightarrow y^{3} = 3 x^{2} y . . . . (i) \bar{} a n d$ $x^{3} - 4 {x y}^{2} - 2 y^{3} = 0 . . . . (i i)$ $(i) \Rightarrow y = 0 o r y^{2} = 3 x^{2}$ $u s i n g a b o v e$ $(i i) \Rightarrow x = 0 o r$ $- 11 x^{3} = 2 y^{3}$ $o r - 11 x = 6 y$ $i n c o n s i s t e n t, s o$ $o n l y s o l u t i o n i s (0, 0) .$
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