On Large Rational Solutions of Cubic Thue Equations (original) (raw)
Related papers
Elliptic Curves and Diophantine Equations
2021
Given an equation of the form f(x, y) = 0, where f is a polynomial in two variables with rational coefficients of degree lower or equal to three, we will study the properties of the set of its rational solutions. We will show that if f is irreducible and the degree of f is three, then the corresponding cubic curve is birationally equivalent to a special cubic curve, often called elliptic. Furthermore, we will define a group law on the set of rational points of an elliptic curve and finish with the proof of Nagell-Lutz theorem, which states that all rational points of finite order in such defined group have integral coordinates.
On cubic Thue equations and the indices of algebraic integers in cubic fields
Applicable Analysis and Discrete Mathematics
Let F(x; y) = ax3 + bx2y + cxy2 + dy3 ? Z[x,y] be an irreducible cubic form. In this paper, we investigate arithmetic properties of the common indices of algebraic integers in cubic fields. For each integer k such that v2(k)??0 (mod 3) and 2v2(-2b3 - 27a2d + 9abc) = 3v2(b2 - 3ac), we prove that the cubic Thue equation F(x,y) = k has no solution (x,y) ? Z2. As application, we construct parametrized families of twisted elliptic curves E : ax3 + bx2 + cx + d = ey2 without integer points (x,y).
Thue's equation as a tool to solve two different problems
Acta et Commentationes Universitatis Tartuensis de Mathematica, 2021
A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.
A parametric family of cubic Thue equations
Journal of Number Theory, 2004
In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we find all solutions to the Thue equation F n ðx; yÞ ¼ x 3 À ðn 3 À 2n 2 þ 3n À 3Þx 2 y À n 2 xy 2 À y 3 ¼ 71;
On the Diophantine equations X^3+Y^3+Z^3+aU^k=a_0U_0^t_0+...+a_nU_n^t_n, k=3,4
2017
In this paper, elliptic curves theory is used for solving the Diophantine equations X^3+Y^3+Z^3+aU^k=a_0U_0^{t_0}+...+a_nU_n^{t_n}, k=3,4 where n, ti are natural numbers and a, a_i are fixed arbitrary rational numbers. We try to transform each case of the above Diophantine equations to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for each case. We also solve these Diophantine equations for some values of n, a, a_i, t_i, and obtain infinitely many solutions for each case, and show among the other things that how sums of four, five, or more cubics can be written as sums of four, five, or more biquadrates as well as sums of 5th powers, 6th powers and so on.
Journal of Number Theory, 1991
We consider the family of cubic Thue equations x3-nx'y-(n + 1)?ry'-y3 = 1, and we give all its solutions for n > 3.67. 1032.
Diophantine m-tuples and Elliptic Curves - front and back matters
Developments in Mathematics, 2024
This book provides an overview of the main results and problems concerning Diophantine m-tuples, i.e., sets of integers or rationals with the property that the product of any two of them is one less than a square, and their connections with elliptic curves. It presents the contributions of famous mathematicians of the past, like Diophantus, Fermat and Euler, as well as some recent results of the author and his collaborators. The book presents fragments of the history of Diophantine m-tuples, emphasising the connections between Diophantine m-tuples and elliptic curves. It shows how elliptic curves are used to solve some longstanding problems on Diophantine m-tuples, such as the existence of infinite families of rational Diophantine sextuples. On the other hand, rational Diophantine m-tuples are used to construct elliptic curves with interesting Mordell–Weil groups, including curves of record rank with a given torsion group. The book contains concrete algorithms and advice on how to use the software package PARI/GP for solving computational problems relevant to the book's topics. This book is primarily intended for researchers and graduate students in Diophantine equations and elliptic curves. However, it can be of interest to other mathematicians interested in number theory and arithmetic geometry. The prerequisites are on the level of a standard first course in elementary number theory. Background in elliptic curves, Diophantine equations and Diophantine approximations is provided in the book. An interested reader may consult also the recent Number Theory book by the author. The author gave a course based on the preliminary version of this book in the academic year 2021/2022 for PhD students at the University of Zagreb. On the course web page, additional materials, like homework exercises (mostly included in the book in the exercise sections at the end of each chapter), seminar topics and links to relevant software, can be found. The book could be used as a textbook for a specialized graduate course, and it may also be suitable for a second reading supplement reference in any course on Diophantine equations and/or elliptic curves at the graduate or undergraduate level.