On a family of cubic Thue equations with 5 solutions
نویسندگان
چکیده
منابع مشابه
Cubic 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.
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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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In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n ≤ 5× 106 and n ≥ N = 1.191× 1019, with n, n+ 2, n2 + 4 square-free, the Thue equation Φn(x, y) = x 4 − nxy − (n + 2n + 4n+ 2)xy − nxy + y = 1 has no integral solution except the trivial ones: (1, 0), (−1, 0), (0, 1), (0,−1).
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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 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: Acta Arithmetica
سال: 2003
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa109-3-7