Thue-Mahler Equations Research Articles

S-unit equation in two variables and Padé approximations

In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory 243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the [Formula: see text]-unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure as well as the local height, we refine the bound due to Evertse [On equations in [Formula: see text]-units and the Thue-Mahler equation, Invent. Math. 75 (1984) 561–584].

Odd values of the Ramanujan tau function

We prove a number of results regarding odd values of the Ramanujan $$\tau $$ -function. For example, we prove the existence of an effectively computable positive constant $$\kappa $$ such that if $$\tau (n)$$ is odd and $$n \ge 25$$ then either $$\begin{aligned} P(\tau (n)) \; > \; \kappa \cdot \frac{\log \log \log {n}}{\log \log \log \log {n}} \end{aligned}$$ or there exists a prime $$p \mid n$$ with $$\tau (p)=0$$ . Here P(m) denotes the largest prime factor of m. We also solve the equation $$\tau (n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$$ and the equations $$\tau (n)=\pm q^b$$ where $$3\le q < 100$$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey–Hellegouarch elliptic curves.

Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations

AbstractWe show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.

On the Number of Solutions in Algebraic Integers of the Thue–Mahler Equation

On the Number of Solutions in Algebraic Integers of the Thue–Mahler Equation

Bornes effectives pour les solutions des équations enS-unités et des équations de Thue–Mahler

Recently, Bombieri and Bombieri and Cohen have developped a new method in effective diophantine approximation, based on a reworked version of the Thue principle, the Dyson lemma and some geometry of numbers. In the present work, we follow their approach, utilizing however estimates for linear forms in two or three logarithms instead of the Thue principle. This tool allows us to sharpen their results. We apply our main estimates to provide new explicit—and, in a certain sense, improved—upper bounds for the size of the solutions of the S-unit equation and of the Thue–Mahler equation. We also derive a new effective irrationality measure for algebraic numbers and compare our estimates with the recent results of Bugeaud and Győry.

Thue and Thue-Mahler Equations over Rings of Integers

A method is given to solve any Thue-Mahler equation when the coefficients and variables come from the ring of integers of a number field. The method involves using an algorithm for solving an S-unit equation and a method of reducing the number of exponential variables. The method is used to solve the equation X-3-Y-3 root 2=+/-5(n) for X,Y is an element of Z[root 2].

$S$-integral solutions to a Weierstrass equation

On détermine les solutions rationnelles de l’équation diophantienne y 2 =x 3 -228x+848 dont les dénominateurs sont des puissances de 2. On applique une idée de Yuri Bilu, qui évite le recours à des équations de Thue et de Thue-Mahler, et qui permet d’obtenir des équations aux (S-) unités à quatre termes dotées de propriétés spéciales, que l’on résout par la théorie des formes linéaires en logarithmes réels et p-adiques.

The number of solutions of the Thue-Mahler equation.

The number of solutions of the Thue-Mahler equation.

Bounds for the solutions of Thue-Mahler equations and norm form equations

Bounds for the solutions of Thue-Mahler equations and norm form equations

Bounds for the solutions of some diophantine equations in terms of discriminants

AbstractSeveral effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

Solving a specific Thue-Mahler equation

The diophantine equation x 3 − 3 x y 2 − y 3 = ± 3 n 0 17 n 1 19 n 2 {x^3} - 3x{y^2} - {y^3} = \pm {3^{{n_0}}}{17^{{n_1}}}{19^{{n_2}}} is completely solved as follows. First, a large upper bound for the variables is obtained from the theory of linear forms in p-adic and real logarithms of algebraic numbers. Then this bound is reduced to a manageable size by p-adic and real computational diophantine approximation, based on the L 3 {L^3} -algorithm. Finally the complete list of solutions is found in a sieving process. The method is in principle applicable to any Thue-Mahler equation, as the authors will show in a forthcoming paper.

Thue-Mahler equations with a small number of solutions.

Thue-Mahler equations with a small number of solutions.

On equations inS-units and the Thue-Mahler equation

On equations inS-units and the Thue-Mahler equation

Elliptic curves of conductor 11

We determine all elliptic curves defined over Q of conductor 11. Firstly, we reduce the problem to one of solving a diophantine equation, namely a certain Thue-Mahler equation. Then we apply recent sharp inequalities for linear forms in the logarithms of algebraic numbers to bound solutions of that equation. Finally, some straightforward computations yield all solutions of the diophantine equation. Our results are in accordance with the conjecture of Taniyama-Weil for conductor 11.

THE THUE-MAHLER EQUATION IN A RELATIVE FIELD AND APPROXIMATION OF ALGEBRAIC NUMBERS BY ALGEBRAIC NUMBERS

A new estimate of the solutions of the generalized Thue-Mahler equation is derived, which explicitly exhibits the influence of all fundamental parameters of the equation on the magnitude of the solutions. Also, an effective power sharpening is given of Liouville's inequality relating to the approximation of algebraic numbers by algebraic numbers of a fixed field, which includes both Archimedean and non-Archimedean valuations.Bibliography: 19 titles.