Thue Equations Research Articles

The number of solutions to the trinomial Thue equation

The number of solutions to the trinomial Thue equation

On the monogenity of pure quartic relative extensions of {{mathbb {Q}}}(i)

We consider pure quartic relative extensions of the number field {{mathbb {Q}}}(i) of type K={{mathbb {Q}}}(root 4 of {a+bi}), where a,bin {{mathbb {Z}}} and bne 0, such that a+biin {{mathbb {Z}}}[i] is square-free. We describe integral bases of these fields. The index form equation is reduced to a relative cubic Thue equation over {{mathbb {Q}}}(i) and some corresponding quadratic form equations. We consider monogenity of K and relative monogenity of K over {{mathbb {Q}}}(i). We shall show how our former method based on the factors of the index form can be used in the relative case to exclude relative monogenity in some cases.

On families of cubic split Thue equations parametrised by linear recurrence sequences

On families of cubic split Thue equations parametrised by linear recurrence sequences

Quartic index form equations and monogenizations of quartic orders

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.

Effective methods for norm-form equations

While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.

An effective Schmidt’s subspace theorem for arbitrary hypersurfaces over function fields

We establish an effective version of Schmidt’s subspace theorem on a smooth projective variety [Formula: see text] over function fields of characteristic zero for arbitrary family of hypersurfaces. We also show an application to study a general Thue equation over function fields.

Calculating “small” solutions of inhomogeneous relative Thue inequalities

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the “small” solutions of these equations. Under “small” solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain “totally real” relative Thue equations can be reduced to absolute Thue equations (equations over $\mathbb{Z}$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating “small” solutions (say with sizes $\leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the “totally real” case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

A classical approach to a parametric family of simultaneous Pell equations with applications to a family of Thue equations

A classical approach to a parametric family of simultaneous Pell equations with applications to a family of Thue equations

On a cubic family of Thue equations involving Fibonacci numbers and powers of two

In this paper we completely solve the family of parametrised Thue equations where Fn is the n-th Fibonacci number. In particular, for any integer n ≥ 3 the Thue equation has only the trivial solutions (±1; 0); (0;∓1);∓(Fn ; 1);∓(2 n ; 1).

PENCILS OF NORM FORM EQUATIONS AND A CONJECTURE OF THOMAS

We introduce a new method to deal with families of norm form equations. These generalize the Thue equations studied first by Thomas using Baker's Method (which, however, we do not use here). We show that for all large integer values of the parameter t, every solution over Z arises from specializing a solution over Z [ T ] by T = t . The results are completely effective.

A positive proportion of locally soluble quartic Thue equations are globally insoluble

AbstractFor any fixed nonzero integer h, we show that a positive proportion of integral binary quartic forms F do locally everywhere represent h, but do not globally represent h. We order classes of integral binary quartic forms by the two generators of their ring of ${\rm GL}_{2}({\mathbb Z})$ -invariants, classically denoted by I and J.

Solutions of a class of quartic Thue inequalities

Let c be an integer such that \(c\ge 3\). In this paper, we use the method of Tzanakis to transform the quartic Thue equation $$\begin{aligned} x^4 +4(c^2-1) x^3y +(8c^2+6) x^2 y^2 +4(c^2-1) xy^3 +y^4 = \mu \end{aligned}$$into a systems of Pell equations. Then, we determine all positive integer solutions (x, y) with \(0<|\mu |\le 2c\).

Thue's equation as a tool to solve two different problems

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.

Thue equations that simultaneously fail the Hasse principle

AbstractWe refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x, y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree ≥ 3 and are ordered by the absolute value of the maximum of the coefficients.

Fourier coefficients of level 1 Hecke eigenforms

Lehmer's 1947 conjecture on whether $\tau(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $\tau(n)$. For example, among the odd numbers $\alpha$ such that $|\alpha|<99$, we determine that $$\tau(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $\tau(n) \neq -|\alpha|$ and that $\tau(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $\Delta E_4 = \sum_{n = 1}^{\infty} \tau_{16}(n)q^n$, we show that \begin{align*}\tau_{16}(n) \notin &\{\pm \ell: 1\leq \ell \leq 99, \ell \text{ is odd}, \ell \neq 33,55,59,67,73,83,89,91\} & \quad \quad \quad \quad \quad \cup \{-33,-55,-59,-67,-89,-91\}.\end{align*} Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros' algorithm for solving hyperelliptic equations, and the theory of continued fractions.

Variations of Lehmer's Conjecture for Ramanujan's tau-function

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for n>1 we prove thatτ(n)∉{±1,±3,±5,±7,±691}. This result is an example of general theorems (see Theorems 1.2 and 1.3 of [2]) for newforms with trivial mod 2 residual Galois representation. Ramanujan's well-known congruences for τ(n) allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty–Coleman method for hyperelliptic curves, and facts about certain Thue equations.

Diagonalizable quartic Thue equations with negative discriminant

The Thue–Siegel method is applied to derive an upper bound for the number of solutions to Thue’s equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used to handle forms whose discriminant is small i

CALCULATING RELATIVE POWER INTEGRAL BASES IN TOTALLY COMPLEX QUARTIC EXTENSIONS OF TOTALLY REAL FIELDS

Some time ago we extended our monogenity investigations and calculations of generators of power integral bases to the relative case. Up to now we considered (usually totally real) extensions of complex quartic fields. In the present paper we consider power integral bases in relative extensions of totally real fields. Totally complex quartic extensions of totally real number fields seems to be the most simple case, that we detail here. As we shall see, even in this case we have to overcome several unexpected difficulties, which we can, however solve by properly (but not trivially) adjusting standard methods. We demonstrate our general algorithm on an explicit example. We describe how the general methods for solving relative index form equations in quartic relative extensions are modified in this case. As a byproduct we show that relative Thue equations in totally complex extensions of totally real fields can only have small solutions, and we construct a special method for the enumeration of small solutions of special unit equations. These statements can be applied to other diophantine problems, as well.

On a class of quartic Thue equations with three parameters

Let k,m and n be integers. In this paper, for a fixed integer μ≠0, we show that the family of Thue equationx4−kmnx3y+(km2−kn2+2)x2y2+kmnxy3+y4=μ,is reducible by Tzanakis's method into a system of pellian equationskV2−(km2+4)U2=−4μ,kZ2−(kn2−4)U2=4μ,with any triple of integers (k,m,n) such that k>0, |n|≥2, |m|≥2. We consider this system for any even integer k≠2□, μ=1 and we prove that for all integers |n|≥2 and |m|≥2 that are sufficiently large and have sufficiently large common divisor this system has only the trivial solutions (U,V,Z,)=(±1,±m,±n). We also show that if k≠2□ is even, then the system has in general at most 8 solutions in positive integers.

Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. 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.