prove-that-x-N-2x-1-2-2x-2-2x-2-is-a-proper-square-number-

Question Number 8022 by Nayon last updated on 28/Sep/16

p r o v e t h a t \forall x \in N

{(2 x - 1)}^{2} + {(2 x^{2} - 2 x)}^{2} i s a p r o p e r

s q u a r e n u m b e r .

Commented by prakash jain last updated on 28/Sep/16

let u = {(2 x - 1)}^{2} + {(2 x^{2} - 2 x)}^{2}

= {(2 x^{2} - 2 x)}^{2} + 4 x^{2} - 4 x + 1

= {(2 x^{2} - 2 x)}^{2} + 2 (2 x^{2} - 2 x) + 1

= {(2 x^{2} - 2 x + 1)}^{2}

x \in N \Rightarrow 2 x^{2} - 2 x + 1 \in N

o r \sqrt{u} \in N

Answered by prakash jain last updated on 02/Oct/16

answer in comments