Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm  

作　　者：Moumouni Djassibo Woba Moumouni Djassibo Woba(Training and Research Unit/Sciences and Technology, University of Ouahigouya, Mra, Burkina Faso)

机构地区：[1]Training and Research Unit/Sciences and Technology, University of Ouahigouya, Mra, Burkina Faso

出　　处：《Applied Mathematics》2024年第9期614-629,共16页应用数学（英文）

摘　　要：Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand 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+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, 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+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand 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+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, 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+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.

关 键 词：Quadratic Form Cornacchia Algorithm Associated Polynomials Euclid’s Algorithm Prime Number 

分 类 号：O17[理学—数学]