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Algebraic independence of the values of certain infinite products and their derivatives related to Fibonacci and Lucas numbers (Analytic Number Theory : Number Theory through Approximation and Asymptotics)openaire
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RATIONAL APPROXIMATIONS TO ALGEBRAIC NUMBERS
Mathematika, 1955This important paper contains a proof of the long conjectured theorem: ``If \(\alpha\) is an algebraic irrational number, and if there are infinitely many fractions \(h/q\) with \(\vert \alpha - h/q\vert \le q^{-\kappa}\), \(q > 0\), \((h, q) = 1\), then \(\kappa\le 2\).'' Much of the proof runs on classical lines.
Davenport, Harold, Roth, Klaus F.
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On constructing Greek ladders to approximate any real algebraic number
International Journal of Mathematical Education in Science and Technology, 2021In 2005, Osler et al. showed that algebraic irrational numbers of the form kn for n,k∈N can be approximated using specific Greek ladders (Osler, T. J., Wright, M., & Orchard, M. (2005).
Jeff Rushall +2 more
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RATIONAL APPROXIMATIONS TO ALGEBRAIC NUMBERS
Mathematics of the USSR-Izvestiya, 1971In this article we derive a new effective estimate of rational approximations to algebraic numbers simultaneously in an Archimedian and several non-Archimedian metrics.
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Rational approximations to algebraic numbers
Mathematika, 1957It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.
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Simultaneous rational approximations to certain algebraic numbers
Mathematical Proceedings of the Cambridge Philosophical Society, 1967It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions ...
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On the Number of Good Simultaneous Approximations to Algebraic Numbers
1991The present work is a continuation of [2] by J. Mueller and the author, where approximations to a single number had been considered. Again it will be convenient to begin with approximations to real numbers in general.
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Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1
Journal für die reine und angewandte Mathematik (Crelles Journal), 2001The main tool of this long and important work is the \textit{multidimensional hypergeometric method} for rational and algebraic approximation of certain algebraic numbers, which was considered first by K.~Mahler and sharpened by G.~Chudnovsky. Here the author obtains explicit results. His most spectacular result is the following definitive Theorem. Let
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2019
Summary: Extending his work in Part I, Mahler now shows that the number of representations of a rational integer \(g\) by a binary form \(F(x,y)\) is at most \(O(|g|^{\varepsilon})\), where \(\varepsilon\) is any arbitrarily small positive constant. Reprint of the author's paper [Math. Ann. 108, 37--55 (1933; Zbl 0006.15604; JFM 39.0269.01)].
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Summary: Extending his work in Part I, Mahler now shows that the number of representations of a rational integer \(g\) by a binary form \(F(x,y)\) is at most \(O(|g|^{\varepsilon})\), where \(\varepsilon\) is any arbitrarily small positive constant. Reprint of the author's paper [Math. Ann. 108, 37--55 (1933; Zbl 0006.15604; JFM 39.0269.01)].
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