Joseph Liouville obtained the first lower bound for the approximation of algebraic numbers: If x is an irrational algebraic number of degree n over the rational numbers, then there exists a constant such that holds for all integers p and q where . This result allowed him to produce the first proven example of a transcendental number, the Liouville constant which does not satisfy Liouville's theorem, whichever degree n is chosen. This link between Diophantine approximations and transcendental number theory continues to the present day. Many of the proof techniques are shared between the two areas. Approximation of algebraic numbers, Thue–Siegel–Roth theorem Over more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound enables us to prove that more numbers are transcendental. The main improvements are due to , , , and , leading finally to the Thue–Siegel–Roth theorem: If is an irrational algebraic number and a (small) positive real number, then there exists a positive constant such that holds for every integer and such that . In some sense, this result is optimal, as the theorem would be false with ε = 0. This is an immediate consequence of the upper bounds described below. Simultaneous approximations of algebraic numbers Subsequently, Wolfgang M. Schmidt generalized this to the case of simultaneous approximations, proving that: If are algebraic numbers such that are linearly independent over the rational numbers and is any given positive real number, then there are only finitely many rational -tuples such that Again, this result is optimal in the sense that one may not remove from the exponent. Effective bounds All preceding lower bounds are not effective, in the sense that the proofs do not provide any way to compute the constant implied in the statements. This means that one cannot use the results or their proofs to obtain bounds on the size of solutions of related Diophantine equations. However, these techniques and results can often be used to bound the number of solutions of such equations. Nevertheless, a refinement of Baker's theorem by Feldman provides an effective bound: if x is an algebraic number of degree n over the rational numbers, then there exist effectively computable constants c(x) > 0 and 0 < d(x) < n such that holds for all rational integers. However, as for every effective version of Baker's theorem, the constants d and 1/c are so large