On The Diophantine Equations X~2-Dy~2=-1 And X~2-Dy~2=4

In this paper,using only the St?rmer theorem and its generalizations on Pell's equation and fundamental properties of Lehmer sequence and the associated Lehmer sequence,we discuss the Diophantine equationsx~2-Dy~2=-1andx~2-Dy~2=4.We obtain the relation between a positive integer solution(x,y)of the Diophantine equation x~2-Dy~2=-1 and its fundamental solution if there is exactly one or two prime divisors of y not dividing D.We obtain the relation between a positive integer solution(x,y)of the Diophantinne equation x~2-Dy~2=4 and its fundamental solution if there is exactly two prime divisors of y not dividing D.And the results are applied to some concrete examples that meet the conditions of the theorem.