Editing Diophantine approximation (section)

=== Effective bounds ===
All preceding lower bounds are not [[effective results in number theory|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'')&nbsp;>&nbsp;0 and 0&nbsp;<&nbsp;''d''(''x'')&nbsp;<&nbsp;''n'' such that

:<math>\left| x- \frac{p}{q} \right|>\frac{c(x)}{|q|^{d(x)}} </math>

holds for all rational integers.

However, as for every effective version of Baker's theorem, the constants ''d'' and 1/''c'' are so large that this effective result cannot be used in practice.