On the System of Diophantine Equations x2 − 6y2 = −5 and x = az2 − b

Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions (x, y, z) of the system of Diophantine equations x (2) - 6y (2) = -5 and x = 2z (2) - 1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equations x (2) - 6y (2) = -5 and x = az (2) - b for each pair of integral parameters a, b. The proof utilizes algebraic number theory and p-adic analysis which successfully avoid discussing the class number and factoring the ideals.