Thue Equations Research Articles

Random Thue and Fermat equations

We consider Thue equations of the form $ax^k+by^k = 1$, and assuming the truth of the $abc$-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat

A family of Thue equations involving powers of units of the simplest cubic 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 ${\mathbf Q}(\lambda)$ of D. Shanks, $\lambda$ 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 $\lambda$ 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 two parameters $n$ and $a$.

Calculating Power Integral bases by Solving Relative Thue Equations

Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

Effective results for Diophantine equations over finitely generated domains

Let $A$ be an arbitrary integral domain of characteristic $0$ that is finitely generated over $\mathbb {Z}$. We consider Thue equations $F(x,y)=\delta $ in $x,y\in A$, where $F$ is a binary form with coefficients from $A$, and $\delta $ is a non-zero elem

Tetranomial Thue equations

Let F ( x , y ) be an irreducible binary form of degree n ⩾ 6 with exactly four nonzero terms. Assuming certain conditions relating the coefficients and the degrees of the different terms are satisfied, we prove upper bounds on the number of equivalent pairs of nontrivial solutions of the Thue equation | F ( x , y ) | = 1 . Improved bounds are provided for a variety of cases, where more information about the form is known.

PERFECT POWERS WITH THREE DIGITS

We solve the equation xa + xb + 1 = yq in positive integers x, y, a, b and q with a > b and q ≥ 2 coprime to φ(x). This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and p-adic logarithms, the hypergeometric method of Thue and Siegel applied p-adically, local methods, and the algorithmic resolution of Thue equations.

On the simplest sextic fields and related Thue equations

We consider the parametric family of sextic Thue equations $$x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda$$ where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

Simplifying method for algebraic approximation of certain algebraic numbers

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters tx 4 − 4sx 3 y − 6tx 2 y 2 +4 sxy 3 + ty 4 = N, where s> 32t 3 . As an application, Ljunggren's equation is solved in an elementary way. DOI: 10.1134/S0001434612090118

UPPER BOUNDS FOR THE NUMBER OF SOLUTIONS TO QUARTIC THUE EQUATIONS

We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve ϕ(x, y) that allows us to use the theory of linear forms in logarithms. This paper improves the results of the author's earlier work with Okazaki [The quartic Thue equations, J. Number Theory130(1) (2010) 40–60] by giving special treatments to forms with respect to their signature.

Binomial Thue equations, ternary equations and power values of polynomials

We explicitly solve the equation Ax n − By n = ±1 and, along the way, we obtain new results for a collection of equations Ax n − By n = z m with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p α q β with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m ⩾ − 1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations X 3 − m X 2 Y − ( m + 3 ) X Y 2 − Y 3 = λ where λ > 0 is a divisor of m 2 + 3 m + 9 and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazakiʼs result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors λ of m 2 + 3 m + 9 . As a consequence, we obtain another proof of Okazakiʼs 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 .

Cubic Thue Equations with Many Solutions

We shall prove that if F is a cubic binary form with integer coefficients and nonzero discriminant then there is a positive number c, which depends on F, such that the Thue equation F(x, y) = m has at least c(log m) 1/2 solutions in integers x and y for infinitely many positive integers m.

The algebraic structure of the set of solutions to the Thue equation

Let F n be a binary form with integral coefficients of degree n ⩾ 2 , let d denote the greatest common divisor of all non-zero coefficients of F n , and let h ⩾ 2 be an integer. We prove that if d = 1 then the Thue equation ( T) F n ( x , y ) = h has relatively few solutions: if A is a subset of the set T ( F n , h ) of all solutions to ( T), with r : = card ( A ) ⩾ n + 1 , then (#) h divides the number Δ ( A ) : = ∏ 1 ⩽ k < l ⩽ r δ ( ξ k , ξ l ) , where ξ k = 〈 x k , y k 〉 ∈ A , 1 ⩽ k ⩽ r , and δ ( ξ k , ξ l ) = x k y l − x l y k . As a corollary we obtain that if h is a prime number then, under weak assumptions on F n , there is a partition of T ( F n , h ) into at most n subsets maximal with respect to condition ( # ) .

On Thue Equations of Splitting Type over Function Fields

In this paper we consider Thue equations of splitting type over the ring k[T ], i.e. they have the form X(X − p1Y ) · · · (X − pd−1Y )− Y d = ξ, with p1, . . . , pd−1 ∈ k[T ] and ξ ∈ k. In particular we show that such Thue equations have only trivial solutions provided the degree of pd−1 is large, with respect to the degree of the other parameters p1, . . . , pd−2.

A Note on the Field Isomorphism Problem of X3+sX+s and Related Cubic Thue Equations

We study the field isomorphism problem of cubic generic polynomial X3+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 x3−2mx2y−9mxy2−m(2m+27)y3=λ where λ2 is a divisor of m3(4m+27)5.

On a family of Thue equations of degree 16

We consider a parameterized family of Thue equations of degree 16 16 . By reducing this family to a system of Pell equations and linear relations, we are able to solve this family.

Thomas’ Family of Thue Equations over Imaginary Quadratic Fields, II

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Cubic Thue equations

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.

Quartic Thue equations

We will give upper bounds upon the number of integral solutions to binary quartic Thue equations. We will also study the geometric properties of a specific family of binary quartic Thue equations to establish sharper upper bounds.

On the solutions of a parametric family of cubic Thue equations

In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we use Baker’s method find all solutions to the Thue equation $$ \begin{gathered} \Phi _n (x,y) = x^3 - n(n^2 + n + 3)(n^2 + 2)x^2 y - (n^3 + 2n^2 + 3n + 3)xy^2 - y^3 = \pm 1, \hfill \\ for n \geqslant 0 \hfill \\ \end{gathered} $$ .