In mathematics, a **Thue equation** is a Diophantine equation of the form

*ƒ*(*x*,*y*) =*r*,

where *ƒ* is an irreducible bivariate form of degree at least 3 over the rational numbers, and *r* is a nonzero rational number. It is named after Axel Thue who in 1909 proved a theorem, now called **Thue's theorem**, that a Thue equation has finitely many solutions in integers *x* and *y*.^{ [1] }

The Thue equation is solvable effectively: there is an explicit bound on the solutions *x*, *y* of the form where constants *C*_{1} and *C*_{2} depend only on the form *ƒ*. A stronger result holds, that if *K* is the field generated by the roots of *ƒ* then the equation has only finitely many solutions with *x* and *y* integers of *K* and again these may be effectively determined.^{ [2] }

Solving a Thue equation can be described as an algorithm^{ [3] } ready for implementation in software. In particular, it is implemented in the following computer algebra systems:

- in PARI/GP as functions
*thueinit()*and*thue()*. - in Magma computer algebra system as functions
*ThueObject()*and*ThueSolve()*. - in Mathematica through
*Reduce*

In mathematics, a **Diophantine equation** is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A **linear Diophantine equation** equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant. An **exponential Diophantine equation** is one in which exponents on terms can be unknowns.

In mathematics, an **equation** is a statement that asserts the equality of two expressions. The word *equation* and its cognates in other languages may have subtly different meanings; for example, in French an *équation* is defined as containing one or more variables, while in English any equality is an equation.

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, *x*, is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

**Hilbert's tenth problem** is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.

For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation.

In number theory, the study of **Diophantine approximation** deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

In mathematics, an **algebraic equation** or **polynomial equation** is an equation of the form

In mathematics, Helmut Hasse's **local–global principle**, also known as the **Hasse principle**, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the *p*-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers *and* in the *p*-adic numbers for each prime *p*.

In mathematics, **Diophantine geometry** is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and *p*-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry.

In mathematics, to **solve an equation** is to find its **solutions**, which are the values that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as *unknowns*. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a **root** of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.

In mathematics, **Roth's theorem** is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of *very good* here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).

**Transcendental number theory** is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.

In number theory and algebraic geometry, a **rational point** of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.

In mathematics, **Siegel's theorem on integral points** states that for a smooth algebraic curve *C* of genus *g* defined over a number field *K*, presented in affine space in a given coordinate system, there are only finitely many points on *C* with coordinates in the ring of integers *O* of *K*, provided *g* > 0.

In mathematics, the **subspace theorem** says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt (1972).

In transcendental number theory and Diophantine approximation, **Siegel's lemma** refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue; Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929. It is a pure existence theorem for a system of linear equations.

In mathematics, in the field of algebraic number theory, an ** S-unit** generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for

In number theory, **Fermat's Last Theorem** states that no three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} for any integer value of *n* greater than 2. The cases *n* = 1 and *n* = 2 have been known since antiquity to have infinitely many solutions.

A **system of polynomial equations** is a set of simultaneous equations *f*_{1} = 0, ..., *f*_{h} = 0 where the *f*_{i} are polynomials in several variables, say *x*_{1}, ..., *x*_{n}, over some field *k*.

- ↑ A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen".
*Journal für die reine und angewandte Mathematik*.**1909**(135): 284–305. doi:10.1515/crll.1909.135.284. - ↑ Baker, Alan (1975).
*Transcendental Number Theory*. Cambridge University Press. p. 38. ISBN 0-521-20461-5. - ↑ N. Tzanakis and B. M. M. de Weger (1989). "On the practical solution of the Thue equation".
*Journal of Number Theory*.**31**(2): 99–132. doi: 10.1016/0022-314X(89)90014-0 .

- Baker, Alan; Wüstholz, Gisbert (2007).
*Logarithmic Forms and Diophantine Geometry*. New Mathematical Monographs.**9**. Cambridge University Press. ISBN 978-0-521-88268-2.

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