| Stormer has proved that if(x,y)is positive integer solution of the Pell equation x~2-Dy~2=±1(D>0,and it is a non-square integer)and all prime factors of y are divide D,(x,y)is the basic solution of the Pell equation x~2-Dy~2=±1.In this paper,the Stormer theorem is extended to the equation kx~2-ly~2=1(k,l∈N,k>1,(k,l)=1,kl is not a square)and kx~2-ly~2=2(k,l∈N,(k,l)=1,kl is not a square),and we discuss a positive integer solution(x,y)of the equation kx~2-ly~2=1,2,and the relationship between the positive integer solution and the minimal solution of the equation kx~2-ly~2=1,2,when x contains one or two prime factors that do not divide k,or y contains one or two prime factors that do not divide l.And the generalized Stormer theorem is applied to obtain some concrete examples that meet the conditions of the theorems. |
