Question

Number of values of $n\in Z$ for which $n^2+n+2$ is a perfect square is

Answer 1

$n^2+n+2=m^2,m\in W$
$n^2+n+(2-m^2)=0$
$D=1-4(2-m^2)=p^2,p\in W$
$4m^2-p^2=7$
$(2m-p)(2m+p)=7$
$2m-p=1$ $\underset{\underset{–}{}}{2m+p=7}4m=8\Rightarrow m=2$
After putting $m=2$ in the equation we get,
$n^2+n-2=0$
$(n+2)(n-1)=0$
$n=1,-2$