- You might be looking for Thue–Siegel–Roth theorem. In Mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in Diophantine approximation to Algebraic
Thue's theorem is a mathematical theorem first proved by Axel Thue in 1909. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Axel Thue ( 19 February 1863 – 7 March 1922) was a Norwegian Mathematician, known for highly original work in Diophantine [1]
It also refers to the 2-dimensional analog of Kepler's conjecture: the regular hexagonal packing is the densest sphere packing in the plane (1890). In Mathematics, the Kepler conjecture is a Conjecture about Sphere packing in three-dimensional Euclidean space.
Statement of Thue's theorem
If f is a bivariate form with rational coefficients which is irreducible over the rational numbers and has degree greater than or equal to 3, and r is a rational number other than 0, then the equation
- f(x,y)=r
has only finitely many solutions in integers x and y. In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same In Mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given set In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions
An equation of the form f(x,y)=r, where f and r are as in the theorem, is called a Thue equation. Hence the theorem can be stated concisely as: Any Thue equation has only finitely many integer solutions.
References
- ^ A. Thue, Über Annäherungswerte algebraischer Zahlen, Journal für die reine und angewandte Mathematik, 135, pages 284-305 (1909)
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