Research on the Number of Solutions to a Special Type of Diophantine Equation (ax−1)(by−1) = 2z2

Abstract

Let b be an odd number. By using elementary methods, we prove that: (1) When x is an odd number and y is an even number, the Diophantine equation (2x−1)(by−1)=2z2 has no positive integer solution except when b is two special types of odd number. (2) When x is an odd number and b≡±3(mod8), the Diophantine equation (2x−1)(by−1)=2z2 has no positive integer solution except where b=3 and is another special type of the odd number.