A parametric family of cubic Thue equations
نویسندگان
چکیده
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 کاملA parametric family of quintic Thue equations
For an integral parameter t ∈ Z we investigate the family of Thue equations F (x, y) = x + (t− 1)xy − (2t + 4t + 4)xy + (t + t + 2t + 4t − 3)xy + (t + t + 5t+ 3)xy + y = ±1 , originating from Emma Lehmer’s family of quintic fields, and show that for |t| ≥ 3.28 ·1015 the only solutions are the trivial ones with x = 0 or y = 0. Our arguments contain some new ideas in comparison with the standard ...
متن کاملOn Correspondence between Solutions of a Parametric Family of Cubic Thue Equations and Non-isomorphic Simplest Cubic Fields
We give a correspondence between non-trivial solutions to a parametric family of cubic Thue equations X − mXY − (m + 3)XY 2 − Y 3 = k where k | m + 3m+ 9 and non-isomorphic simplest cubic fields. By applying R. Okazaki’s result for non-isomorphic simplest cubic fields, we obtain all solutions to the family of cubic Thue equations for k | m + 3m+ 9.
متن کاملOn Correspondence between Solutions of a Parametric Family of Cubic Thue Equations and Isomorphic Simplest Cubic Fields
We give a correspondence between non-trivial solutions to a parametric family of cubic Thue equations X − mXY − (m + 3)XY 2 − Y 3 = k where k | m+3m+9 and isomorphic simplest cubic fields. By applying R. Okazaki’s result for isomorphic simplest cubic fields, we obtain all solutions to the family of cubic Thue equations for k | m + 3m + 9.
متن کاملOn two-parametric family of quartic Thue equations
We show that for all integers m and n there are no non-trivial solutions of Thue equation x − 2mnxy + 2 ( m − n + 1 ) xy + 2mnxy + y = 1, satisfying the additional condition gcd(xy,mn) = 1.
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2004
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2004.01.009