Thue Equations Research Articles

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.

A positive proportion of Thue equations fail the integral Hasse principle

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.

On cubic Thue equations and the indices of algebraic integers in cubic fields

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).

A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences

In 1927, Artin conjectured that any integer other than −1 or a perfect square generates the multiplicative group (Z/pZ)× for infinitely many p. In 2000, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture. In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the first N terms of a non-degenerate binary recurrence sequence. We also prove a weaker version of the same theorem, and give three proofs that we consider to be of independent interest. The first proof uses a transcendence result of Stewart, the second uses a theorem of Bombieri and Schmidt on Thue equations and the third uses Mumford's gap principle for counting points on curves by their height. We finally prove a disjunction theorem, where we consider the set of primes satisfying either our two-variable condition or the original condition of Artin's conjecture. We give an unconditional lower bound for the number of such primes.

Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype goes at least back to Mordell, which associates a pair of binary forms to an integral point on an elliptic curve. Other ingredients of the proof include the upper bounds for the number of solutions of a Thue equation by Evertse and Akhtari-Okazaki, and the estimation of the number of binary quartic forms by Bhargava-Shankar. Our method applies to $S$-integral points to some extent, although our present knowledge is insufficient to deduce an upper bound for the average number of them. We work out the numerical example with $S=\{2\}$.

DETERMINING ELEMENTS OF MINIMAL INDEX IN AN INFINITE FAMILY OF TOTALLY REAL BICYCLIC BIQUADRATIC NUMBER FIELDS

Let c ≠ 2 be a positive integer such that c and c + 4 are square-free. We consider the infinite parametric family of bicyclic biquadratic fields K = Q(sqrt {; ; 2c}; ; , sqrt {; ; 2(c + 4)}; ; ). We determine the integral basis of the field. We show that K admits no power integral basis, determine the minimal index and all elements of minimal index. We use the solutions of a parametric family of quartic Thue equations and extensive numerical calculations by Maple and Magma are also involved.

On equation Xn − 1 = BZn

We consider the Diophantine equation [Formula: see text], where [Formula: see text] is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, [Formula: see text], has a common divisor with the exponent [Formula: see text], or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell–Ljunggren equation: [Formula: see text]. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution. The first such family was proved by Bennett in his seminal paper on binary Thue equations (see [M. A. Bennet, Rational approximation to algebraic numbers of small height: the Diophantine equation [Formula: see text], J. Reine Angew. Math. 535 (2001) 1–49]).

Fundamental units for a family of totally real cubic orders and the diophantine equation u(u + a)(u + 2a) = v(v + 1)

We determine a system of fundamental units of the totally real cubic orders [Formula: see text], where [Formula: see text] is a root of [Formula: see text], [Formula: see text], which arise in our study here of the equation [Formula: see text]. For potentially infinitely many values of [Formula: see text], we reduce the resolution of [Formula: see text] to the resolution of the Thue equations of the form [Formula: see text]. In particular, we solve [Formula: see text] for [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] among [Formula: see text]. This extends and corrects the work of Godinho, Porto and Togbé for [Formula: see text], [Formula: see text], as well as extending earlier work of Mordell for [Formula: see text].

FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD

Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $\sigma_1,\sigma_2,\dots,\sigma_d$ the embeddings of $K$ into $\mathbb{C}$, $\alpha$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-\sigma_i(\alpha) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-\sigma_i(\alpha\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effective upper bound for the solutions $(x,y,a)\in{\mathbb{Z}}^3$ of the Diophantine inequalities $$ 0<|F_a(x,y)|\le m $$ for which $xy\not=0$ and ${\mathbb{Q}}(\alpha \upsilon^a)=K$. Our estimate involves an effectively computable constant depending only on $d$; it is explicit in terms of $m$, in terms of the heights of $F_0$ and of $\upsilon$, and in terms of the regulator of the number field $K$.

Complete solutions to a family of Thue equations of degree 12

Complete solutions to a family of Thue equations of degree 12

Thue equations and CM-fields

We obtain a polynomial-type upper bounds for the size and the number of the integral solutions of Thue equations \(F(X,Y) = b\) defined over a totally real number field K, assuming that F(X, 1) has a root \(\alpha \) such that \(K(\alpha )\) is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.

Elementary resolution of a family of quartic Thue equations over function fields

We consider and completely solve the parametrized family of Thue equations $$\begin{aligned} X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi , \end{aligned}$$ where the solutions x, y come from the ring $$\mathbb {C}[T]$$ , the parameter $$\lambda \in \mathbb {C}[T]$$ is some non-constant polynomial and $$0\ne \xi \in \mathbb {C}$$ . It is a function field analogue of the family solved by Mignotte, Pethő and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are very elementary (in particular compared to previous results on parametrized Thue equations over function fields).

REPRESENTATION OF SMALL INTEGERS BY BINARY FORMS

We establish some upper bounds for the number of integer solutions to the Thue inequality |$|F(x , y)| \leq m$|⁠, where |$F$| is a binary form of degree |$n \geq 3$| and with non-zero discriminant |$D$|⁠, and |$m$| is an integer. Our upper bounds are independent of |$m$|⁠, when |$m$| is smaller than |$|D|^{{1}/{4(n-1)}}$|⁠. We also consider the Thue equation |$|F(x , y)| = m$| and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer |$m$|⁠, when |$m < |D|^{{1}/{2(n-1)}}$|⁠.

On simultaneous Pell equations and related Thue equations

In this paper, we prove that the simultaneous Pell equations \[ x 2 − ( m 2 − 1 ) y 2 = 1 , z 2 − ( n 2 − 1 ) y 2 = 1 x^2-(m^2-1)y^2=1,\, z^2-(n^2-1)y^2=1 \] have only a positive integer solution ( x , y , z ) = ( m , 1 , n ) (x, y, z) = (m, 1, n) if m &gt; n ≤ m + m ε , 0 &gt; ε &gt; 1 m &gt; n \le m+m^{\varepsilon }, 0 &gt; \varepsilon &gt; 1 and m ≥ 202304 1 1 − ε m \ge 202304^{\frac {1}{1-\varepsilon }} . Using a computational reduction method we can omit the lower bound for m m when m &gt; n ≤ m 1 5 m&gt;n\le m^{\frac {1}{5}} . Moreover, we apply our main result to a family of Thue equations in two parameters studied by Jadrijević.

Calculating “Small” Solutions of Relative Thue Equations

Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions. On the other hand, to compute all solutions (i.e., to prove the nonexistence of large solutions) is a time-consuming procedure. Therefore, it is useful to have a fast algorithm to calculate the “small” solutions, especially if “small” means less than, e.g., 10100. Such an algorithm was constructed by A. Pethö in 1987 based on continued fractions.In the present paper, we construct a similar type of fast algorithm to calculate “small” solutions of relative Thue equations. Our method is based on the LLL reduction algorithm. We illustrate the method with explicit examples. The algorithm has several applications.

English

We consider binomial Thue equations of type $x^n-my^n=\pm 1$ in $x,y\in Z$. Optimizing the method of Peth\H o we perform an extensive calculation by a high performance computer to determine all solutions with $\max(|x|,|y|)<10^{500}$ of binomial Thue equations for $m<10^7$ for exponents $n=3,4,5,7,11,13,17,19,23,29$.

Erratum to: “On the solutions of a parametric family of cubic Thue equations” [Bull Braz Math Soc, New Series 39 (4) (2008), 537–554

In our original paper [4], Theorem 2.1 of [2], that we quoted, was not properly proved. This was done by Lee-Louboutin [3]. The gap doesn’t affect the final result of the paper.

Thue equations and lattices

We consider Diophantine equations of the kind $|F(x,y)|= m$, where $F(X,Y )\in \bz [X,Y]$ is a homogeneous polynomial of degree $d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of Stewart and provide heuristics for a conjecture of Stewart.

Mordell’s equation: a classical approach

We solve the Diophantine equation$Y^{2}=X^{3}+k$for all nonzero integers$k$with$|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

Extremal families of cubic Thue equations

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.