On the Diophantine equation 2m+nx2=yn

In this note, we prove that the Diophantine equation 2m+nx2=yn in positive integers x, y, m, n has the only solution (x,y,m,n)=(21,11,3,3) with n>1 and gcd(nx,y)=1. In fact, for n=3,15, we transform the above equation into several elliptic curves for which we determine all their {2}-integer points. For n≠3,15, we apply the result of Yu.F. Bilu, G. Hanrot and P.M. Voutier about primitive divisors of Lehmer sequences.