Thue Equations Research Articles

On the representation of unity by binary cubic forms

If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) = 1 possesses at most ten solutions in integers x and y, nine if F has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z[x, y], then this equation has at most 2 solutions, unless F (x, y) is equivalent under GL2(Z)action to either x(x2 − xy − y2) or x(x2 − 2y2). The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations F (x, y) = 1 for F cubic and irreducible of positive discriminant DF ≤ 106. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form F (x, y) = m and to Mordell’s equation y2 = x3 + k, where m and k are nonzero integers.

Computing Power Integral Bases in Quartic Relative Extensions

We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (I. Gaál, A. Pethő, and M. Pohst, 1993, J. Symbolic Comput.16, 563–584; 1996, J. Number Theory57, 90–104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations determine the generators of power integral bases of K over M up to translation by integers of M and multiplication by unit factors of M. The new method is based on our ability to solve relative Thue equations efficiently by the algorithm in (I. Gaál and M. Pohst, 2000, Math. Comp., to appear). In the case K is an octic field with a quadratic subfield M we can also consider the absolute index of elements of K, having relative index 1 over M. In order to determine all generators of power integral bases of K (over Q) we determine the corresponding translating elements and unit factors properly. This is done by solving an equation similar to an inhomogeneous Thue equation. We illustrate our algorithms with detailed examples.

A Parametric Family of Quintic Thue Equations II

In this paper we completely solve the family of Thue equations where is an integral parameter. In particular, for , the only solutions are the trivial ones with x = 0 or y = 0. The result is achieved by sharpening the estimates of part I of the paper and by solving Thue equations with the method of Bilu and Hanrot.

Solving Index Form Equations in Fields of Degree 9 with Cubic Subfields

We describe an efficient algorithm for solving index form equations in number fields of degree 9 which are composites of cubic fields with coprime discriminants. We develop the algorithm in detail for the case of complex cubic fields, but the main steps of the procedure are also applicable for other cases. Our most important tool is the main theorem of a recent paper of Gaál (1998a). In view of this result the index form equation in the ninth degree field implies relative index form equations over the subfields. In our case these equations are cubic relative Thue equations over cubic fields. The main purpose of the paper is to show that this approach is much more efficient than the direct method, which consists of reducing the index form equation to unit equations over the normal closure of the original field. At the end of the paper we describe our computational experience. Many ideas of the paper can be applied to develop fast algorithms for solving index form equations in other types of higher degree fields which are composites of subfields.

Counting solutions to trinomial Thue equations: a different approach

We consider the problem of counting solutions to a trinomial Thue equation — that is, an equation ( ∗ ) | F ( x , y ) | = 1 , \begin{equation*} |F(x,y)| = 1,\tag {$*$} \end{equation*} where F F is an irreducible form in Z [ x , y ] Z[x,y] with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the “Thue-Siegel principle" and its relation to ( ∗ ) (*) . In this paper we give specific numerical bounds for the number of solutions to ( ∗ ) (*) by a somewhat different approach, the difference lying in the initial step — solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.

Thue Equations Associated with Ankeny-Brauer-Chowla Number Fields

For a wide class of one-parameter families of Thue equations of arbitrary degree n [ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

A parametric family of quintic Thue equations

For an integral parameter t ∈ Z we investigate the family of Thue equations F(x,y) = x 5 + (t - 1) 2 x 4 y - (2t 3 + 4t + 4)x 3 y 2 + (t 4 + t 3 + 2t 2 + 4t - 3)x 2 y 3 + (t 3 + t 2 + 5t + 3)xy 4 + y 5 = ±1, originating from Emma Lehmer's family of quintic fields, and show that for |t| > 3.28.10 15 the only solutions are the trivial ones with x= 0 or y = 0. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

Solving Thue equations without the full unit group

The main problem when solving a Thue equation is the computation of the unit group of a certain number field. In this paper we show that the knowledge of a subgroup of finite index is actually sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given.

On the solutions of a family of quartic Thue equations

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 × 10 6 and n ≥ N = 1.191 x 10 19 , with n, n + 2, n 2 + 4 square-free, the Thue equation Φ n (x,y) = x 4 - n 2 x 3 y - (n 3 + 2n 2 + 4n + 2)x 2 y 2 - n 2 xy 3 + y 4 = 1 has no integral solution except the trivial ones: (1, 0), (-1,0), (0,1), (0, -1).

On Families of Parametrized Thue Equations

For the family of parametrized Thue equations(X+aY)(X−d2Y)…(X−dn−1Y)(X−aY)−Yn=±1,wheren⩾4,didistinct integers satisfying ∑di≠0 or ∏di≠0, all solutions are determined for sufficiently large values of the integral parameterausing bounds on linear forms in logarithms.

Thue equations with composite fields

We consider the Thue equation $F(x,y)=a$, where $F$ is an irreducible form of degree $n\geq 3$.We describe a method of resolution which takes advantage of the fact that the number field generated by a root of $F(1,y)$ has small subfields. We illustrate this method by solving several real cyclotomic equations of degrees as large as 2505. || Considerons l'equation de Thue $F(x,y)=a$, avec $F$ une forme irreductible homogene de degre $n\geq 3$. Nous decrivons une methode de resolution permettant de tirer profit de l'existence de petits sous-corps du corps de nombres engendre par une racine

Effective solution of families of Thue equations containing several parameters

F (X,Y ) = m, where F ∈ Z[X,Y ] is an irreducible form of degree n ≥ 3 and m 6= 0 a fixed integer, has only finitely many solutions. However, this proof is non-effective and does not give any bounds for the size of the possible solutions. In 1968, A. Baker could give effective bounds based on his famous theory on linear forms in logarithms of algebraic numbers. In the last decades, this method was refined (see for instance Baker and Wustholz [1] and Waldschmidt [29]). Recent explicit upper bounds for the solutions of Thue equations have been given by Bugeaud and Győry [3]. Algorithms for the solution of a single Thue equation have been developed by several authors (see Bilu and Hanrot [2]). E. Thomas [25] was the first to deal with a parametrized family of Thue equations; since then, some families have been investigated: cubic families have been discussed by Mignotte [16], Lee [12], and Mignotte and Tzanakis [19], a cubic inequality has been solved by Mignotte, Pethő, and Lemmermeyer [17]; quartic families have been considered by Pethő [20], Mignotte, Pethő, and Roth [18], Lettl and Pethő [13], Chen and Voutier [4], and Heuberger, Pethő, and Tichy [10]; Wakabayashi [28] dealt with a quartic inequality, Pethő and Tichy [21] solved a two-parametric quartic family, quintic families have been investigated by Heuberger [9] and Gaal and Lettl [6], a sextic family has been solved by Lettl, Pethő, and Voutier [14, 15]; see

On a conjecture of Siegel

Letk be an algebraically closed field of characteristic 0 and letf(x, y)∈k[t][x,y] be a polynomial in two variables with coefficients ink[t]. One is interested in solving the equationf(x,y)=0 with polynomialsx,y∈k[t]. Two solutions(x,y), (x′, y′) areproportional ifx′/x andy′/y are non-zero constants ink and a solution(x,y) isprimitive if the polynomialsx andy have no common root. The main result of this paper is that for a certain class of polynomialsf, which includes Thue equations with sufficiently lacunary exponents, the number of non-proportional, primitive solutions is bounded solely in terms of the number of monomials $$a_i (t)x^{\alpha _1 } y^{\beta _i } $$ appearing in the polynomialf(x,y). This verifies the analogue of a conjecture of Siegel for this class of polynomials. The proof is an application of theabc-theorem in function fields to certain determinantal varieties arising from the elimination of the coefficients of the polynomialf(x,y), together with an inductive argument on the numberr of monomials inf(x,y).

Différence entre puissances et carrés d'entiers

We solve the equationx2+7=ynfor fixed values ofn⩽5000 using the classical machinery for Thue equation and modular tools. Next we solve it for fixed values ofy⩽109using linear forms of logarithms and reduction techniques.

On Two-parametric Quartic Families of Diophantine Problems

Two-parametric quartic Thue equations are completely solved for sufficiently large values of the parameters. Exceptional units are computed in related quartic number fields. The method depends heavily on A. Baker's theory of linear forms in logarithms and symbolic computation in MAPLE.

On a Family of Quintic Thue Equations

We consider the parametrized quintic family of Thue equationsX(X2−Y2) (X2−a2Y2)−Y5=± 1 a∈Zand prove that it only has trivial solutions for ∣a∣≥ 3.6× 1019using a recent estimate for linear forms in three logarithms of algebraic numbers by P. Voutier.

Solving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms

We determine the solutions in integers of the equation y2 = (x + p)(x2 + p2) for p = 167, 223, 337, 1201. The method used was suggested to us by Yu. Bilu, and is shown to be in some cases more efficient than other general purpose methods for solving such equations, namely the elliptic logarithms method and the use of Thue equations.

Complete Solution of the Diophantine EquationX2+1=dY4and a Related Family of Quartic Thue Equations

In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the title. We prove that this equation has at most one solution in positive integers whend⩾3. Moreover, when such a solution exists, it is of the form(u, v)where (u, v) is the fundamental solution ofX2+1=dY2.

Solving Thue Equations of High Degree

We propose a general method for numerical solution of Thue equations, which allows one to solve in reasonable time Thue equations of high degree (provided necessary algebraic number theory data is available). We illustrate our method, solving completely concrete Thue equations of degrees 19 and 33.

Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields

LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions of[formula]can be presented by finitely many expressions of the formx=fx(p, q),y=fy(p, q),z=fz(p, q), wherefx, fy, fz∈Z[P, Q] are binary quadratic forms andp, qare coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is irreducible it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutionsp, qof that equation then yield all solutionsx, y, zof (1). We also discuss two applications. In [8] we showed that the problem of solving index form equations in quartic numbers fieldsKcan be reduced to the resolution of a cubic equationF(u, v)=iand a corresponding system of quadratic equationsQ1(x, y, z)=u,Q2(x, y, z)=v, whereFis a binary cubic form andQ1, Q2are ternary quadratic forms. In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic fieldK. The second application is to the calculation of all integral points of an elliptic curve.