On cubic Thue equations and the indices of algebraic integers in cubic fields
نویسندگان
چکیده
منابع مشابه
Integer points on cubic Thue equations
We prove that there are infinitely many inequivalent cubic binary forms F with content 1 for which the Thue equation F (x, y) = m has ≫ (logm) solutions in integers x and y for infinitely many integers m.
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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.
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We shall prove that if F is a cubic binary form with integer coefficients and non-zero discriminant then there is a positive number c, which depends on F, such that the Thue equation F (x, y) = m has at least c(logm) solutions in integers x and y for infinitely many positive integers m.
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 Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell
In 1657, French lawyer and amateur mathematician Pierre de Fermat became interested in positive integer solutions u and v to the equation u − 61 v = 1. He posed a sadistic challenge to established mathematicians of the day, such as the Englishman William Brouncker and John Wallis, asking if they could find the solutions he found – without telling them the answer, of course. See, Fermat had foun...
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ژورنال
عنوان ژورنال: Applicable Analysis and Discrete Mathematics
سال: 2019
ISSN: 1452-8630,2406-100X
DOI: 10.2298/aadm190608032b