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.
متن کاملCubic Thue equations with many solutions
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 کاملFamilies of Cubic Thue Equations with Effective Bounds for the Solutions
To each non totally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.
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...
متن کاملOn Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell
The paper is largely expository. The first part is devoted to studying integer solutions to Pell’s Equation: u2 − dv2 = 1. The authors present the classic construction of a fundamental solution via continued fractions, from which all solutions can be derived. The primary focus of the second part is on rational solutions to the Thue’s equation, u3−dv3 = 1. The authors explain why these rational ...
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ژورنال
عنوان ژورنال: Journal de Théorie des Nombres de Bordeaux
سال: 2015
ISSN: 1246-7405,2118-8572
DOI: 10.5802/jtnb.907