On Two-Parametric Quartic Families of Diophantine Problems
نویسندگان
چکیده
Let n ≥ 3, v1(x), . . . , vn−1(x) ∈ Z[x] and u ∈ {−1, 1}, then x(x− v1(a)y) · · · (x− vn−1(a)y) + uy = ±1 is called a parametrized familiy of Thue equations, if a ∈ Z and the solutions x, y are searched in Z; cf. Thomas (1993). There are several results concerning parametrized families of cubic and quartic families of Thue equations, see Mignotte et al. (1996) and the references therein. Thomas (1993) proved that if 0 < b < c and a ≥ [2 × 10(b + 2c)] 4.85 c−b , then the only solutions of the equation x(x− ay)(x− ay) + uy = ±1 u = ±1 are ±{(1, 0), (0, u), (au, u), (au, u)}. As far as we know this is the only case, when a two-parametric familiy of Thue equations is completely solved. In this paper we consider diophantine problems which are related to a two-parametric quartic polynomial. Let 1 < a < b, a, b ∈ Z and Pa,b(x) = x(x− 1)(x− a)(x− b)− 1. This polynomial is irreducible (see Halter-Koch et al. (1998)). Denoting by α one of the zeros of Pa,b(x) the number field K = Q(α) is quartic. Let O = Z[α] and UO be the group of units of O. In the first part, we study the algebraic properties of K. We prove
منابع مشابه
A parametric family of quartic Thue equations
In this paper we prove that the Diophantine equation x − 4cxy + (6c+ 2)xy + 4cxy + y = 1, where c ≥ 3 is an integer, has only the trivial solutions (±1, 0), (0,±1). Using the method of Tzanakis, we show that solving this quartic Thue equation reduces to solving the system of Pellian equations (2c+ 1)U − 2cV 2 = 1, (c− 2)U − cZ = −2, and we prove that all solutions of this system are given by (U...
متن کاملUnits in Some Parametric Families of Quartic Fields
In this article we compute fundamental units for three parametric families of number fields of degree 4 with unit rank 2 and 3 generated by polynomials with Galois group D4 and S4.
متن کاملB-SPLINE METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEMS
In this work the collocation method based on quartic B-spline is developed and applied to two-point boundary value problem in ordinary diferential equations. The error analysis and convergence of presented method is discussed. The method illustrated by two test examples which verify that the presented method is applicable and considerable accurate.
متن کاملSolving index form equations in two parametric families of biquadratic fields∗
In this paper we find a minimal index and determine all integral elements with the minimal index in two families of totally real bicyclic biquadratic fields Kc = Q (√ (c− 2) c, √ (c+ 2) c ) and Lc = Q (√ (c− 2) c, √ (c+ 4) c ) . AMS subject classifications: Primary 11D57, 11A55; Secondary 11B37, 11J68, 11J86, 11Y50
متن کاملA Generalization of a Theorem of Bumby on Quartic Diophantine Equations
Given a parametrized family of cubic models of elliptic curves Et over Q, it is a notoriously difficult problem to find absolute bounds for the number of integral points on Et (and, indeed, in many cases, it is unlikely such bounds even exist). Perhaps somewhat surprisingly, the situation is often radically different for quartic models. In a series of classical papers, Ljunggren (see e.g. [5] a...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Symb. Comput.
دوره 26 شماره
صفحات -
تاریخ انتشار 1998