Michael A. Bennett, Amir Ghadermarzi
We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction.
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In this paper, we investigate the common index divisors of cyclic cubic fields. Let $a,b,c,d$ and $k$ are integers, we then solve the following Thue cubic equations:: \[ax^3+bx^2y+cxy^2+dy^3= k\ \] when $a,bc+d$ are odd and $3$ doesn't divide $v_2(k)$.
We revisit a work by R. Okazaki and prove that for every cubic binary form F(x, y) with large enough discriminant, the Thue equation |F(x, y)| = 1 has at most 7 solutions in integers x and y.
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.
Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number. We completely solve the family of cubic Thue equations $${(X-F_nY)(X-L_nY)X-Y^3=\pm1}$$ and show that there are no non-trivial solutions for $n\neq 1,3$.
We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided that h is smaller than D^{1/4}.
In this paper we completely solve the family of parametrised Thue equations \[ X(X-F_n Y)(X-2^n Y)-Y^3=\pm 1, \] where $F_n$ is the $n$-th Fibonacci number. In particular, for any integer $n\geq 3$ the Thue equation has only the trivial solutions $(\pm 1,0), (0,\mp 1), \mp(F_n,1), \mp(2^n,1)$.
Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=λ\] where $λ>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $λ$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389…
We study the field isomorphism problem of cubic generic polynomial $X^3+sX+s$ over the field of rational numbers with the specialization of the parameter $s$ to nonzero rational integers $m$ via primitive solutions to the family of cubic Thue equations $x^3-2mx^2y-9mxy^2-m(2m+27)y^3=λ$ where $λ^2$ is a divisor of $m^3(4m+27)^5$.
Let $(A_n)_{n\in \mathbb{N}}, (B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations \[ |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1) \}$, if the parameter $n$ is larger than some effectively computable constant.
The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(λ)$ of D. Shanks, $λ$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1) $ of $λ$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the …
For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.
In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we u…
Let $h(x,y)$ be a non-degenerate binary cubic form with integral coefficients, and let $S$ be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers $x,y$ such that $h(x,y)$ is an $S$-unit. In the present paper, we reverse a well known argument, which seems to go back to Shafarevich, and use the modularity of elliptic curves over $\mathbb{Q}$ to give upper bounds for the number of solutions of such a Thue-Mahler equation. In addition, our methods gives an effective method for determining all solutions, and we …
Complete Resolution Of A Family Of Twisted Thue Equations
本文求解了Lévesque与Waldschmidt提出的一类扭曲Thue方程，找出其所有整数解，解答了相关公开问题。
Power-saving bounds for Thue--Mahler and Mordell equations
该研究针对三次Thue--Mahler方程证明了对regulator具节能依赖的有效界，将Mordell方程整数解的Stark界指数降至1/2+ε，改进了相关分离下界并在Frey高度猜想的导体方面取得节能进展。
Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.
We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that the cardinality of the symmetric difference of any two elements $F,F'\in\mathcal{F}$ is not a multiple of 4. We prove that the maximal size of $\mathcal{F}$ is bounded by $2n$, unless $n\equiv{}3\mod4$ when it is bounded by $2n+2$. Our method uses cubic forms on $\mathbb{F}_2^n$ and the Hurwitz-Radon theory of square identities. We also apply this theory to obtain some information about boolean cubic forms and so-called additive quadruples.
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