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Erschienen in:
2016 | OriginalPaper | Buchkapitel
Linear Forms in Logarithms
verfasst von : Sanda Bujačić, Alan Filipin
Erschienen in: Diophantine Analysis
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Abstract
Hilbert’s problems form a list of twenty-three problems in mathematics published by David Hilbert, a German mathematician, in 1900. The problems were all unsolved at the time and several of them were very influential for 20th century mathematics. Hilbert believed it was essential for mathematicians to find new machineries and methods in order to solve the mentioned problems. The seventh problem deals with the transcendence of \(\alpha ^\beta \) for algebraic \(\alpha \ne 0,1\) and irrational algebraic \(\beta \). This problem was solved by Gelfond and (independently) Schneider. In 1935, Gelfond found a lower bound for the absolute value of the linear form He proved that where \(h(\varLambda )\) is logarithmic height of the linear form \(\varLambda \), \(\kappa >5\) and \(\gg \) denotes inequality that is valid up to an unspecified constant factor. He noticed that generalization of his results could prove a huge amount of unsolved problems in number theory.
$$\varLambda =\beta _1\log \alpha _1+\beta _2\log \alpha _2\ne 0.$$
$$\log |\varLambda |\gg -h(\varLambda )^\kappa ,$$
In 1966 and 1967, in his papers “Linear forms in logarithms of algebraic numbers I, II, III”, A. Baker gave an effective lower bound on the absolute value of a nonzero linear form in logarithms of algebraic numbers, that is, for a nonzero expression of the form where \(\alpha _1, \dots , \alpha _n\) are algebraic numbers and \(b_1, \dots , b_n\) are integers.
$$\sum _{i=1}^{n}b_i\log \alpha _i,$$
In these notes, we introduce definitions and theorems that are crucial for understanding and applications of linear forms in logarithms. Some Baker type inequalities that are easy to apply are introduced. In order to illustrate this very important machinery, we present some examples and show, among other things, that the largest Fibonacci number having only one repeated digit in its decimal expression is 55, that \(d=120\) is the only positive integer such that the set \(\{d+1, 3d+1, 8d+1\}\) consists of all perfect squares and that some parametric families of \(D(-1)\)-triples cannot be extended to \(D(-1)\)-quadruples.