If x∈Z ∧y non-negative integer such that x^2 +10x+23=2^y find out x,y


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Question Number 217952 by dscm last updated on 23/Mar/25

$I f x \in Z \land y n o n - n e g a t i v e i n t e g e r$ $s u c h t h a t$ $x^{2} + 10 x + 23 = 2^{y}$ $f i n d o u t x, y$
	Answered by Frix last updated on 23/Mar/25

	$x = - 5 \pm \sqrt{2^{y} + 2}$ $\Rightarrow \sqrt{2^{y} + 2} \in N \Rightarrow y = 1 \Rightarrow x = - 7 \lor x = - 3$
	Commented by dscm last updated on 23/Mar/25

	$T h a n k s s i r, b u t w h a t i s t h e g u a r a n t e e$ $t h a t y = 1 i s o n l y v a l u e f o r 2^{y} + 2$ $t o b e p e r f e c t s q u a r e .$
	Answered by Ghisom last updated on 23/Mar/25

	$x = t - 5$ $t^{2} - 2 = 2^{y}$ $\Rightarrow t = 2 n$ $4 n^{2} - 2 = 2^{y}$ $2 n^{2} = 2^{y - 1} + 1$ $\Rightarrow 2 ∣ (2^{y - 1} + 1) \Rightarrow y = 1 \Rightarrow n = \pm 1 \Rightarrow t = \pm 2 \Rightarrow$ $x = \pm 2 - 5$ $x = - 7 \lor x = - 3$
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