the implied constant can be made explicit. The 'Siegel period' Many of the principal results of analytic number theory that were proved in the period 1900–1950 were in fact ineffective. The main examples were: The Thue–Siegel–Roth theorem Siegel's theorem on integral points, from 1929 The 1934 theorem of Hans Heilbronn and Edward Linfoot on the class number 1 problem The 1935 result on the Siegel zero The Siegel–Walfisz theorem based on the Siegel zero. The concrete information that was left theoretically incomplete included lower bounds for class numbers (ideal class groups for some families of number fields grow); and bounds for the best rational approximations to algebraic numbers in terms of denominators. These latter could be read quite directly as results on Diophantine equations, after the work of Axel Thue. The result used for Liouville numbers in the proof is effective in the way it applies the mean value theorem: but improvements (to what is now the Thue–Siegel–Roth theorem) were not. Later work Later results, particularly of Alan Baker, changed the position. Qualitatively speaking, Baker's theorems look weaker, but they have explicit constants and can actually be applied, in conjunction with machine computation, to prove that lists of solutions (suspected to be complete) are actually the entire solution set. Theoretical issues The difficulties here were met by radically different proof techniques, taking much more care about proofs by contradiction. The logic involved is closer to proof theory than to that of computability theory and computable functions. It is rather loosely conjectured that the difficulties may lie in the realm of computational complexity theory. Ineffective results are still being proved in the shape A or B, where we have no way of telling which. References External links Analytic number theory Diophantine equations