The Exponential Diophantine Equation 2x+by=cz

Let b and c be fixed coprime odd positive integers with min{b,c}>1. In this paper, a classification of all positive integer solutions (x,y,z) of the equation 2x+by=cz is given. Further, by an elementary approach, we prove that if c=b+2, then the equation has only the positive integer solution (x,y,z)=(1,1,1), except for (b,x,y,z)=(89,13,1,2) and (2r-1,r+2,2,2), where r is a positive integer with r≥2