On the Diophantine equation NX2+2L3M=YNNX2+2L3M=YN

We prove that the Diophantine equation NX2+2L3M=YNNX2+2L3M=YN has no solutions (N,X,Y,L,M)(N,X,Y,L,M) in positive integers with N>1N>1 and gcd(NX,Y)=1gcd(NX,Y)=1, generalizing results of Luca, Wang and Wang, and Luca and Soydan. Our proofs use results of Bilu, Hanrot, and Voutier on defective Lehmer pairs.