Thue–Siegel–Roth theorem
The set of Thue -Siegel -Roth from the theory of Diophantine approximations in number theory was proved by Klaus Friedrich Roth, following preparatory work of Axel Thue and Carl Ludwig Siegel 1955.
He states that for every algebraic number and any inequality ( p, q relatively prime )
(Inequality 1)
Only finitely many solutions has. Previously, Joseph Liouville already had shown in 1844 that for irrational in
(Inequality 2)
Applies. Here, n is the degree of the algebraic equation with root. Elementary considerations also show that (see below). Axel Thue showed in 1908 that Carl Ludwig Siegel and 1921 in his dissertation that. So Roth improved on.
By making this a finite number of solutions aside, can be calculated from (inequality 1) follows that for sufficiently large q for each irrational:
(Inequality 3)
With only one dependent of C. This is the " best" possible such set because after Peter Gustav Lejeune Dirichlet ( Dirichlet's approximation theorem ) any real number approximant p / q has to be closer than (there is even an infinite number, eg. the approximants of the continued fraction representations of these numbers).
The proof of Roth indicates no method to find such solutions or C limit. That would be interesting to learn about the number of solutions of Diophantine equations (that is, integer or rational solutions of algebraic equations, for example, the one real root in ( inequality 3) ). Such effective methods were introduced in the 1960s by Alan Baker in the theory of transcendental numbers and Diophantine equations.