Cornachia’s algorithm can be adapted to the case of the equation x2 + dy2 = n and even to the case of ax2 + bxy + cy2 = n . For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2 + y2 = n ). Starting from a quadratic form with two variables f (x, y) = ax2 + bxy + cy2 and n an integer. We have shown that a primitive positive solution (u,v) of the equation f (x, y) = n is admissible if it is obtained in the following way: we take α modulo n such that f (α ,1) ≡ 0mod n , u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn D ) (possibly α itself) and the equation f (x, y) = n . has an integer solution u in y . At the end of our work, it also appears that the Cornacchia algorithm is good for the form n = ax2 + bxy + cy2 if all the primitive positive integer solutions of the equation f (x, y) = n are admissible, i.e. computable by the algorithmic process.
Loading....