Solutions and solvabilities of general binary quadric indefinite equations

Abstract

To this day for special m and small I c I , solvable, c are obtained about general equations s2 my2 = c with positive squarefree integer m and integer c by Ankeny, Mollin et uL . Afterwards, Zhang in ref. [ I ] greatly developed their correspondent results and gave all proper integer solutions and solvable c about the equations for general m . Then Lu in ref. [2] gave results for small I q I about general binary quadric indefinite equations ax2 bxy cy2 = q (1 1 with positive square-free integer d = b2 + 4uc and integer a. b. c . q . Here we will spread their results about solutions and solvabilities of eq. ( 1 ) for general q . Based on the classification theory of binary quadric formL3], we may think I b I < a < I c 1 to eq. ( I ) , where g. c. d. ( a . b . c . ) = 1. Suppose that the simple continued fraction of a = b+G -. is a = [ a o , al, ..., a k ] with period k and that a,' = P, + f i 2a 2 Q, , P,, Q, E Z , Pn Lao, al, ..., a,] = -(g.c.d(p, , q,) = 1 ) . Ref. [2 ] gets, ior Po= b and Q o = a , Qn