On the resolution of simultaneous Pell equations ∗
نویسندگان
چکیده
We descibe an alternative procedure for solving automatically simultaneous Pell equations with relatively small coefficients. The word “automatically” means to indicate that the algorithm can be implemented in Magma. Numerous famous examples are verified and a new theorem is proved by running simply the corresponding Magma procedure requires only the six coefficients of the system a1x 2 + b1y 2 = c1, a2x 2 + b2z 2 = c2.
منابع مشابه
Simultaneous Pell equations
Let R and S be positive integers with R < S. We shall call the simultaneous Diophantine equations x −Ry = 1, z − Sy = 1 simultaneous Pell equations in R and S. Each such pair has the trivial solution (1, 0, 1) but some pairs have nontrivial solutions too. For example, if R = 11 and S = 56, then (199, 60, 449) is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible...
متن کاملOn two classes of simultaneous Pell equations with no solutions
In this paper we describe two classes of simultaneous Pell equations of the form x2−dy2 = z2−ey2 = 1 with no solutions in positive integers x, y, z. The proof is elementary and covers the case (d, e) = (8, 5), which was solved by E. Brown using very deep methods.
متن کاملComplete solution of a family of simultaneous Pellian equations
Let ck = P 2 2k + 1, where Pk denotes the k th Pell number. It is proved that for all positive integers k all solutions of the system of simultaneous Pellian equations z − ckx = ck − 1, 2z − cky = ck − 2 are given by (x, y, z) = (0,±1,±P2k). This result implies that there does not exist positive integers d > c > 2 such that the product of any two distinct elements of the set {1, 2, c, d} dimini...
متن کاملOn the sum of Pell and Jacobsthal numbers by matrix method
In this paper, we define two $n$-square upper Hessenberg matrices one of which corresponds to the adjacency matrix of a directed pseudo graph. We investigate relations between permanents and determinants of these upper Hessenberg matrices, and sum formulas of the well-known Pell and Jacobsthal sequences. Finally, we present two Maple 13 procedures in order to calculate permanents of t...
متن کاملPairs of Pell Equations Having at Most One Common Solution in Positive Integers
We prove that, for positive integers m and b, the number of simultaneous solutions in positive integers to x2− (4m2−1)y2 = 1, y2−bz2 = 1 is at most one.
متن کامل