On the Diophantine equation 2m + nx2 = yn

In this note, we prove that the Diophantine equation 2(m) + nx(2) = y(n) 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 not equal 3,15, we apply the result of Yu.F. Bilu, G. Hanrot and P.M. Voutier about primitive divisors of Lehmer sequences. (C) 2012 Elsevier Inc. All rights reserved.