Question

Find n, if $2^{200}-2^{192}31+2^n$ is a perfect square.

Answer 1

Find x, if $2^{200}-\left(2^{192}\right)\times 31+2^n$ is

a perfect square.

$\to 2^{200}-\left(2^{192}\right)\times 31+2^n$

$=2^{192}(2^3-31+2^{n-192})$

taking $2^{192}$ common

let $n-192$ be 'k'

$=2^{192}(256-31+2^k)$

$=2^{192}(225+2^k)$

$2^{192}$ is perfect square, we have to select

k such that $(225+2^k)$ will because

perfect square

we will use trial and error

method and we will stock from $k=1$

$1,2,3,4,5$ will not satisfy

only $(k=6)$ will satisfy

& $k=6$

$\Rightarrow n-192=6$

$n=192+6$

$n=198$