On the Diophantine equation 2m+nx2=yn

Abstract

In this note, we prove that the Diophantine equation 2 m + n x 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 ( n x , 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.