Question

$2^8+2^{11}+2{}^n$ is a perfect square if and only if n = 12.

proove it.

Answer 1

Let $m^2=2^8+2^{11}+2^n$
$2^n=m^2-2^{8}9=m^2-(2^4.3{)}^2$
$=(m-48)(m+48)$
Let $m-48=2^s$ & $m+48=2^t$
where s + t = n
$\therefore$ $m=2^s+48=2^t-48$
$=2^t-2^s=96=2^s(2^{t-s}-1)=96$
$\because$ $2^{t-s}-1$ is odd
$\Rightarrow$ $2^s(2^{t-s}-1)=2^{5}3\Rightarrow s=5,t=7$