engleski

Linear Forms in Logarithms

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 the 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 was dealing with the transcendence of alpha^beta for algebraic alpha≠0, 1 and irrational algebraic beta. This problem was solved separately by Gelfond and Schneider. In 1935, Gelfond found a lower bound for the absolute value of the linear form Lambda=beta_1*log(alpha_1)+beta_2*log(alpha_2)≠ 0. He noticed that generalization of his results could prove a huge amount of unsolved problems in number theory. 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. 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.

Diophantine approximation ; Diophantine equation ; Transcendence theory ; Linear form in logarithms ; Baker's method

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