If-x-Z-y-non-negative-integer-such-that-x-2-10x-23-2-y-find-out-x-y-

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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