# Thue's theorem

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Thue's theorem may refer to the following mathematical theorems named after Axel Thue:

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

Axel Thue, was a Norwegian mathematician, known for highly original work in diophantine approximation, and combinatorics.

• Thue equation has finitely many solutions in integers.
• Thue's lemma, which asserts that every modular integer may be written as a modular fraction with numerator and denominator bounded by the square root of the modulus.
• Thue–Siegel–Roth theorem, also known as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers.
• The 2-dimensional analog of Kepler's conjecture: the regular hexagonal packing is the densest sphere packing in the plane (1890).

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

In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus.

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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

Fermat's little theorem states that if p is a prime number, then for any integer a, the number apa is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even. For example, 6 is even because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of even numbers include −4, 0, 82 and 178. In particular, zero is an even number. Some examples of odd numbers are −5, 3, 29, and 73.

In mathematics, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001.

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, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers. A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic variety V over K; the typical question is about the nature of the set V(K) of points on V with co-ordinates in K, and by means of height functions quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number. Given the geometric approach, the consideration of homogeneous equations and homogeneous co-ordinates is fundamental, for the same reasons that projective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions can be treated in the same way as an affine variety may be considered inside a projective variety that has extra points at infinity.

Carl Ludwig Siegel was a German mathematician specialising in number theory and celestial mechanics. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation and the Siegel mass formula for quadratic forms. He was named as one of the most important mathematicians of the 20th century.

Klaus Friedrich Roth was a German-born British mathematician known for work on diophantine approximation, the large sieve, and irregularities of distribution.

In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number may not have too 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).

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A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

In mathematics, the subspace theorem is a result obtained by Wolfgang M. Schmidt (1972). It states that if L1,...,Ln are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x with

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