Linear forms in the logarithms of algebraic numbers

Abstract

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.