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A method in the theory of [[Diophantine approximations|Diophantine approximations]], created by A. Thue [[#References|[1]]] in connection with the problem of approximating algebraic numbers (cf. [[Algebraic number|Algebraic number]]) by rational numbers: Find a quantity $\nu=\nu(n)$ such that for each algebraic number $\alpha$ of degree $n$ the inequality

$$\left|\alpha-\frac pq\right|<\frac{1}{q^{\nu+\epsilon}}\label{1}\tag{1}$$

has a finite number of solutions in rational integers $p$ and $q$, $q>0$, for any $\epsilon>0$, and an infinite number of solutions for any $\epsilon<0$.

Thue proved that $\nu\leq(n/2)+1$. Thue's method is based on properties of a special polynomial $f(x,y)$ of two variables $x,y$ with integer coefficients, and the hypothesis that there exist two solutions of \eqref{1} for $\nu\leq(n/2)+1$ with sufficiently large values of $q$. Thue's theorem has many important applications in number theory. In particular, it implies that the Diophantine equation

$$F(x,y)=m,\label{2}\tag{2}$$

where $F(x,y)$ is an irreducible form in the variables $x$ and $y$ with integer coefficients and of degree $n\leq3$, while $m$ is an integer, cannot have more than a finite number of solutions in integers $x$ and $y$.

The best possible estimate of the size of $\nu$ in \eqref{1} was obtained by K.F. Roth [[#References|[2]]], by generalizing Thue's method to the case of a polynomial in any number of variables, similar to the polynomial $f(x,y)$, and making use of the large number of solutions of \eqref{1}. The result, called the Thue–Siegel–Roth theorem, states that $\nu=2$ for any $n\geq2$. Thue's method has a generalization to the case of approximation of algebraic numbers by algebraic numbers. Thue's method is a general method for proving the finiteness of the number of integer points on a wide class of curves on algebraic varieties (see [[Diophantine geometry|Diophantine geometry]]; [[Diophantine set|Diophantine set]]). Apart from this, Thue's method has essential deficiencies: it is a non-effective method in the sense that it does not provide an answer to the question whether there exist in fact solutions of the inequalities \eqref{1} or the corresponding equations \eqref{2} that can be made use of in the proofs. Thus, Thue's method, in solving the question on the finiteness of the number of solutions of equation \eqref{2}, does not provide the possibility of determining whether an actual equation of this type is solvable and what the sizes of the estimates of the solutions $x,y$ are in their dependence on $F$.

See also [[Diophantine approximation, problems of effective|Diophantine approximation, problems of effective]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Thue,   "Ueber Annäherungswerte algebraischer Zahlen"  ''J. Reine Angew. Math.'' , '''135'''  (1909)  pp. 284–305</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.F. Roth,   "Rational approximation to algebraic numbers"  ''Mathematika'' , '''2''' :  1  (1955)  pp. 1–20</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> , ''Problems in the theory of Diophantine approximations'' , Moscow  (1974)  (In Russian; translated from English)</TD></TR></table>



====Comments====
Thue's method has been extended by C.L. Siegel to obtain $\nu<2\sqrt n$. For a good exposition, see [[#References|[a1]]]. This method, known as the Thue–Siegel method, has recently met with spectacular success when P. Vojta [[#References|[a2]]] showed how it could be used to give a new proof of the [[Mordell conjecture|Mordell conjecture]]. A considerable and very accessible simplification of this proof has been given by E. Bombieri. For other generalizations see [[Thue–Siegel–Roth theorem|Thue–Siegel–Roth theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Landau,   "Vorlesungen über Zahlentheorie" , Chelsea, reprint  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Vojta,   "Siegel's theorem in the compact case"  ''Ann. of Math.''  (Forthcoming)</TD></TR></table>