A note on the approximation by continued fractions under an extra condition.
Dajani, Karma, Kraaikamp, Cor (1998)
The New York Journal of Mathematics [electronic only]
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Dajani, Karma, Kraaikamp, Cor (1998)
The New York Journal of Mathematics [electronic only]
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Christian Friesen (1991)
Acta Arithmetica
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Iekata Shiokawa (1975)
Publications mathématiques et informatique de Rennes
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Karma Dajani, Cor Kraaikamp (2000)
Colloquium Mathematicae
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We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known...
Yasushige Watase (2015)
Formalized Mathematics
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In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1]. ...
Dominique Barbolosi, Hendrik Jager (1994)
Journal de théorie des nombres de Bordeaux
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K. Dajani, C. Kraaikamp, V. Masarotto (2012)
Acta Arithmetica
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Nair, R. (1998)
The New York Journal of Mathematics [electronic only]
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Barry R. Smith (2015)
Acta Arithmetica
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We show that for a fixed integer n ≠ ±2, the congruence x² + nx ± 1 ≡ 0 (mod α) has the solution β with 0 < β < α if and only if α/β has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number α/β > 1 in lowest terms has a symmetric continued fraction precisely when β² ≡ ±1(mod α ).
Tibor Šalát (1967)
Matematický časopis
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C. Ryll-Nardzewski (1951)
Studia Mathematica
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Yusuf Hartono, Cor Kraaikamp, Fritz Schweiger (2002)
Journal de théorie des nombres de Bordeaux
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In this paper the Engel continued fraction (ECF) expansion of any is introduced. Basic and ergodic properties of this expansion are studied. Also the relation between the ECF and F. Ryde’s (MNK) is studied.
Kentaro Nakaishi (2006)
Acta Arithmetica
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