On the Diophantine equation \(x^2+2=y^n\) (Q1572839)

The author gives an elementary proof of the fact that the only solutions of the Diophantine equation \(x^2+2=y^n\) for \(n>1\) are \((x,y,n)=(\pm 5,3,3)\). The first proof of this result is due to Nagell, but he uses a non elementary result of Mahler on binary quadratic forms. An important tool of the present proof is the curious polynomial identity: \[ \sum_{j=0}^{[k/2]}(-1)^j {k-j \choose j} (XY)^j(X+Y)^{k-2j}=\sum_{d=0}^k X^dY^{k-d}. \]