Diophantine approximation and the equation (a2cxk−1)(b2cyk−1)=(abczk−1)2(a2cxk−1)(b2cyk−1)=(abczk−1)2

We prove that the Diophantine equation (a2cxk−1)(b2cyk−1)=(abczk−1)2(a2cxk−1)(b2cyk−1)=(abczk−1)2 has no solutions in positive integers with x, y  , z>1z>1, k≥7k≥7 and a2xk≠b2yka2xk≠b2yk.